On Approximate Matrix Factorization and TASE W-Methods for the Time Integration of Parabolic Partial Differential Equations

Abstract

Linearly implicit methods for Ordinary Differential Equations combined with the application of Approximate Matrix Factorization (AMF) provide efficient numerical methods for the solution of large semi-discrete parabolic Partial Differential Equations in several spatial dimensions. Interesting particular subclasses of such linearly implicit methods are the so-called W-methods and the TASE W-methods recently introduced in González-Pinto et al. (Appl Numer Math, 188:129–145, 2023) with the aim of reducing the computational cost of the TASE Runge–Kutta methods in Bassenne et al. (J Comput Phys 424:109847, 2021) and Calvo et al. (J Comp Phys 436:110316, 2021). In this paper, we study the application of the AMF approach in combination with TASE W-methods. While for AMF W-methods the temporal order of consistency is immediately obtained from that of the underlying W-method, this property needs a more thorough analysis for the newly introduced AMF-TASE W-methods. For these latter methods it is described which are the additional order conditions to be fulfilled and it is shown that the parallel structure of the methods is crucial to retain the order of consistency of the underlying TASE W-method. Numerical experiments are presented in three spatial dimensions to assess the consistency result and to show that the proposed schemes are competitive with other well-known good performing AMF W-methods.

1 Introduction

For the approximate solution of Ordinary Differential Equations (ODEs), aspects such as the order of consistency, the stability properties or the ease of implementation, among others, are crucial when selecting a numerical integrator. Excellent textbooks on this topic of research are, e.g., [3, 16, 17]. On one hand, explicit methods, such as Explicit Runge–Kutta (RK) methods, despite their simple implementation, are well-known to possess bounded stability regions for linear problems and, consequently, this limits its practical interest for stiff problems. On the other hand, although implicit methods can be constructed with high order of consistency and excellent stability, their computational cost becomes quite demanding specially when the dimension of the current ODE is large. The implementation of implicit methods, in particular for stiff problems, is typically performed through Newton-type iterations which require the computation or approximation for the Jacobian J of the vector field of the ODE under consideration and the solution of large linear systems involving J. An intermediate alternative with similar convergence and stability properties as for implicit methods is given by the class of linearly implicit methods [26], whose implementation only requires the solution of linear systems of the form $(I-\theta \tau W)K=V$, where I is the identity matrix, $\theta >0$ is a given real parameter, $\tau$ is a stepsize and W is a certain matrix which typically approximates the true Jacobian J. In some cases, this matrix W could even be chosen arbitrarily, leading to the subclass of W-methods [25] (see also Subsect. 2.1 below). Here, we consider the case where W has some connection with the true Jacobian at the current time-step. The solution of such linear systems can still be very demanding for large ODEs, specially those arising after the spatial discretization of Partial Differential Equations (PDEs) in several spatial dimensions. In such context, a nice alternative to reduce the computational cost due to the linear algebra is based on using the so-called Approximate Matrix Factorization (AMF), see e.g. [20] and [21, Ch. IV.5], where the matrix $(I-\theta \tau W)$ is replaced by a product $(I-\theta \tau J_1)\cdots (I-\theta \tau J_d)$. Here the matrices $J_1,\ldots ,J_d$ usually correspond to a splitting of the exact Jacobian J and have a much simpler structure (e.g., a banded structure) which allows to reduce the algebra to the level of one-dimensional problems. As a matter of fact, the so-called AMF W-methods have proven to be an efficient choice for the time integration of multidimensional parabolic PDEs (see, e.g., [9, 12, 13]).

Recently, in [2], and subsequently in [5], in the context of stiff ODEs of moderate size, the authors considered the stabilization of explicit s-stage RK methods of order $p=s$ ($2\le s\le 4$) by multiplying the vector field of the ODE by the so-called TASE matrix operator -standing for Time Accurate and highly Stable Explicit- $T_s(\tau J)=\sum _{\ell =1}^s \omega _\ell (I-\theta _\ell \tau J)^{-1}$, where the parameters $\omega _\ell$ are selected in such a way that $T_s(\tau J)=I+\mathcal {O}(\tau ^s)$, $\tau \rightarrow 0,$ thus ensuring that the modified numerical method retains the order of consistency of the underlying explicit RK method. Furthermore, for the resulting TASE-RK methods, the parameters $\theta _\ell >0$ can be chosen providing nearly optimal linear stability properties for one-dimensional problems. This approach has been further investigated in [1] including the option of complex parameters $\theta _\ell$ appearing in conjugate pairs and also in [6] in the context of explicit multistep peer methods. Since TASE-RK methods need s LU decompositions for the matrices $(I-\theta _\ell \tau J)$ per integration step, the authors in [4] have next proposed a singly implicit structure for the TASE operator denoted as STASE and given by $ST_s(\tau J)=\sum _{\ell =1}^s \omega _\ell (I-\theta \tau J)^{-\ell }$, where now only one LU decomposition per integration step is required.

All the mentioned methods share a linearly implicit structure. In [11] the authors observed that such TASE-RK methods can be regarded as particular W-methods with a parallel structure, which led to the definition of TASE W-methods (see Subsect. 2.2 below). This new class of TASE W-method was shown to provide optimal linear stability properties while retaining the order of consistency $p=s$ of the underlying explicit RK method by considering just two terms (instead of s) in the TASE operator. It is then the main goal of this paper to study the applicability and consistency of this latter class of methods on very large ODEs corresponding to semi-discrete PDEs when combined with the AMF approach. While the consistency of AMF W-methods follows immediately from the consistency of the underlying W-method [9], this property is not so evident in the case of the newly defined class of AMF-TASE W-methods since they require different parameters $\theta _\ell$ in the TASE operator. Anyway, it will be shown below that, due to the parallel structure of the methods, an AMF-TASE W-method inherits the order of consistency of its underlying TASE W-method. The proof of this result will require to study order conditions for a new variant of W-methods whose internal stages are defined in terms of different W matrices connected to the AMF.

The rest of the paper is divided into sections as follows. In Sect. 2, for ease of presentation we review in Subsects. 2.1 and 2.2 the definition and order conditions for W-methods [17, p. 114–116] and the TASE W-methods recently introduced in [11], respectively. The goal of Sect. 3 is to incorporate the AMF approach within TASE W-methods so that these latter methods can be applied more efficiently to numerically solve large split ODEs arising from spatial semi-discretization of parabolic PDEs. Therefore, in Sect. 3 we present the new class of AMF-TASE W-methods and the main theoretical result of this manuscript, which states that the AMF retains the order of consistency of the underlying TASE W-method. The proof of this result will be postponed to Sect. 5. Previously, in Sect. 4, we obtain the necessary order conditions corresponding to a more general class of W-methods which employ different W matrices related to the AMF for each internal stage of the method. Section 6 provides a numerical confirmation of the consistency of AMF-TASE W-methods on three 3D semidiscrete parabolic PDEs corresponding to a nonlinear diffusion–reaction problem from combustion theory, a linear advection–diffusion problem including terms with mixed derivatives and a linear diffusion PDE with variable coefficients. The numerical experiments also show that AMF-TASE W-methods are competitive with other AMF W-methods from the literature. In Sect. 7, we report the proofs of the theorems stated in Sects. 4 and 5 to arrive at the main result of the manuscript. Such proofs are not placed immediately after the statements of the theorems for ease of reading and presentation. Finally, some concluding remarks are drawn in Sect. 8.

2 W-Methods and TASE W-Methods

2.1 W-Methods for ODEs

Let us consider Initial Value Problems in ODEs, initially assumed in autonomous form

$$\begin{aligned} y'(t)=f(y), \quad y(t_0)=y_0, \quad t\in [t_0,t^*],\qquad y,f\in \mathbb {R}^M. \end{aligned}$$

(1)

The stage equation and advancing formula of an s-stage W-method for (1) are

$$\begin{aligned} \left\{ \begin{aligned} (I_M-\theta _i\tau W) K_i&= \tau f\Bigl (y_n+ \textstyle \sum _{j=1}^{i-1} a_{i,j} K_{j}\Bigr ) + \sum _{j=1}^{i-1} \ell _{i,j} K_j, \quad i=1,\ldots ,s, \\ y_{n+1}&=y_n + \textstyle \sum _{i=1}^{s} b_i K_i, \end{aligned} \right. \end{aligned}$$

(2)

with stepsize $\tau =t_{n+1}-t_n$ to advance from $(t_n,y_n)$ to $(t_{n+1},y_{n+1})$. Here, $I_q$ denotes the identity matrix of dimension q, whereas the matrix $W\in \mathbb {R}^{M\times M}$ can be chosen arbitrarily, although in practice it is usually chosen such that $W= f'(y_n)+\mathcal {O}(\tau )$. In particular, when $W=f'(y_n)$ the class of Rosenbrock-Wanner (ROW) methods is obtained. The method is characterized by the matrices $A=(a_{i,j})$, $L=(\ell _{i,j})\in \mathbb {R}^{s\times s}$, $b=(b_i)\in \mathbb {R}^{s}$ and $\Theta =Diag(\theta _i)\in \mathbb {R}^{s\times s}$, with A and L being strictly lower triangular and with $\theta _i>0$, $i=1,\ldots ,s.$ In practice, it is typically assumed that $\theta _i=\theta$, for all i, in order to reduce the computational cost when solving the stage equation in (2).

W-methods are competitive to other classes of time integration methods for ODEs not only due to their linearly implicit structure, but also due to the fact that they can provide a high order of consistency and good linear stability properties (such as L-stability when $W=J$). However, W-methods of a given order $p\ge 3$ are difficult to construct due to the high number of order conditions to be met. In particular, under the situation that W is arbitrary, it is well-known that there are not W-methods with order $p\ge s$, when $s\ge 3$ [23, 26], although when $W= f'(y_n)+\mathcal {O}(\tau )$, methods of order $p=s$, with $s=3,4$, can be obtained (see, e.g., [12, 13, 17, 21, 23]).

The derivation of general order conditions for W-methods (2) can be performed in a simpler way by considering the following alternative formulation of the methods

$$\begin{aligned} \left\{ \begin{aligned} (I_M-\theta _i \tau W)\tilde{K}_i&= \tau f\Bigl (y_n+ \textstyle \sum _{j=1}^{i-1}\tilde{a}_{i,j} \tilde{K}_{j}\Bigr ) + \theta _i\tau W \textstyle \sum _{j=1}^{i-1} \tilde{\gamma }_{i,j} \tilde{K}_j, \quad i=1,\ldots ,s, \\ y_{n+1}&=y_n +\textstyle \sum _{i=1}^{s} \tilde{b}_i \tilde{K}_i, \end{aligned} \right. \end{aligned}$$

(3)

which is obtained through the linear change $\tilde{K}:= ((I_s-L)\otimes I_M) K$, where $K=(K_i)_{i=1}^{s}$ and $\tilde{K}=(\tilde{K}_i)_{i=1}^s\in \mathbb {R}^{sM}$, are the stages of (2) and (3), respectively. In this alternative formulation (3), the coefficients of the method then become $\tilde{A}=(\tilde{a}_{i,j})$, ${\Gamma }=( \gamma _{i,j})\in \mathbb {R}^{s\times s}$, $\tilde{b}=(\tilde{b}_i)\in \mathbb {R}^{s}$, with $\tilde{A}$ and $\tilde{\Gamma }$ strictly lower triangular matrices related to the ones in (2) by means of

$$\begin{aligned}&\tilde{A}=A(I_s-L)^{-1}, \quad \tilde{b}^\top =b^\top (I_s-L)^{-1}, \quad \text {and} \end{aligned}$$

(4)

$$\begin{aligned}&\Gamma =\Theta +\Theta \tilde{\Gamma },\quad \text {with}\quad \tilde{\Gamma }=(I_s-L)^{-1}-I_s=(\tilde{\gamma }_{i,j})\in \mathbb {R}^{s\times s}. \end{aligned}$$

(5)

The underlying explicit RK method is obtained by setting $W=0$ in (3), in which case we just write $L=0=\Gamma$ so that $\tilde{A}=A$ and $\tilde{b}=b$. Tables 1 and 2 (from [11]) collect the order conditions up to order 4 for RK methods and W-methods, respectively. Here $\textbf{1}_q:=(1,\ldots ,1)^\top \in \mathbb {R}^q$ and $\varvec{*}$ denotes the componentwise product of vectors in $\mathbb {R}^s$, whereas the powers of a vector are understood componentwise. See [17, p. 114–116] for a consistency theory based on Butcher trees for W-methods of arbitrary order p.

Table 1 Order conditions for RK methods $(\tilde{A}, \tilde{b})$, with $c=\tilde{A}\textbf{1}_s$, up to order 4

Table 2 Order conditions for W-methods $(\tilde{A}, \Gamma , \tilde{b})$ in (3), with $c=\tilde{A}\textbf{1}_s$, up to order 4 (plus the RK order conditions in Table 1)

2.2 TASE W-Methods for ODEs

In [11] the authors introduced the class of s-stage TASE$_r$ W-methods defined as

$$\begin{aligned} \left\{ \begin{aligned} (I_M-\theta _\ell \tau W) K_i^{[\ell ]}&=\tau f\Bigl (y_n+ \textstyle \sum _{j=1}^{i-1} a_{i,j} K_{j} \Bigr )+ \sum _{j=1}^{i-1} \ell _{i,j} K_{j},\quad \ell =1,\ldots ,r,\\ K_i&=\textstyle \sum _{\ell =1}^r \omega _\ell K_i^{[\ell ]},\quad i=1,\ldots ,s,\\ y_{n+1}&=y_n + \textstyle \sum _{i=1}^{s} b_i K_i. \end{aligned}\right. \end{aligned}$$

(6)

As for the methods (2), the matrix W can be arbitrary. The method (6) is now defined by the matrix coefficients $A=(a_{i,j})$, $L=(\ell _{i,j})\in \mathbb {R}^{s\times s}$, $b=(b_i)\in \mathbb {R}^{s}$, $\omega =(\omega _\ell )\in \mathbb {R}^{r}$ and $\Theta =Diag(\theta _\ell )\in \mathbb {R}^{r\times r}$, with A and L strictly lower triangular. Furthermore, $\omega _\ell$, $\ell =1,\ldots ,r,$ are real parameters fulfilling $\sum _{\ell =1}^r \omega _\ell =1$ and $\theta _\ell >0$, $\ell =1,\ldots ,r,$ are pairwise distinct. The stages $K_i^{[\ell ]}$, $\ell =1,\ldots ,r,$ are regarded as parallel stages since for each $i=1,\ldots ,s$ they can be computed in parallel, whereas $K_i$, $i=1,\ldots ,s$, are sequential stages. Note that, per integration step, (6) requires r LU decompositions for the matrices $I_M-\theta _\ell \tau W$, and the solution of $r\cdot s$ linear systems for the parallel stages $K_i^{[\ell ]}$. When $r=1$, W-methods of the form (2) with $\theta _i=\theta$, $i=1,\ldots ,s$, are obtained.

The methods (6) were proposed in [11] to reduce the number of linear systems to be solved per integration step of the so-called TASE-RK (Time Accurate and highly Stable Explicit) methods previously introduced in [2] and [5], which are obtained from (6) when $r=s$ and $L=0$. In this latter situation, the so-called TASE operator

$$\begin{aligned} T_r(\tau W)=\textstyle \sum _{\ell =1}^r \omega _\ell (I_M-\theta _\ell \tau W)^{-1} \end{aligned}$$

(7)

is required to fulfil (when $r=s$) $T_s(\tau W)=I_M+\mathcal {O}(\tau ^s)$. From here, it is immediately observed that the TASE-RK method retains the algebraic order $p=s$ of the underlying explicit RK method (when $s=2,3,4$).

The newly introduced TASE W-methods in [11] have the flexibility that the natural number r in (6) can be chosen such that $r<s$ in order to reduce the computational cost of TASE-RK methods (for which $r=s$ for consistency reasons). In fact, it is observed that (6) can be regarded as an $(r\cdot s)$-stage W-method in the formulation (2) with coefficient matrices

$$\begin{aligned} \begin{array}{c} \widehat{A}:=A\otimes \Omega ,\qquad \widehat{L}:=L\otimes \Omega ,\qquad \widehat{b}\;^\top :=b^\top \otimes \omega ^\top ,\\ \Theta := Diag(\theta _\ell )_{\ell =1}^r,\qquad \Omega =\begin{bmatrix} \omega _1 &{} \ldots &{} \omega _r\\ &{} \vdots &{} \\ \omega _1 &{} \ldots &{} \omega _r \end{bmatrix}_{r\times r}=\textbf{1}_{r} \;\omega ^\top ,\qquad \omega =\begin{bmatrix}\omega _1\\ \vdots \\ \omega _r\end{bmatrix}. \end{array} \end{aligned}$$

(8)

Observe that, since $\sum _{\ell =1}^r \omega _\ell =1$, it holds that

$$\begin{aligned} \Omega ^k=\Omega \; (k\ge 2),\quad \Omega \textbf{1}_r=\textbf{1}_r\quad \text {and}\quad \omega ^\top \Omega =\omega ^\top . \end{aligned}$$

(9)

Alternatively, in formulation (3), the coefficients of a TASE W-method are given by

$$\begin{aligned} \begin{array}{lll} \widehat{\tilde{A}}:=\tilde{A}\otimes \Omega ,&{}\quad \widehat{\tilde{b}}^\top :=\tilde{b}^\top \otimes \omega ^\top ,&{}\quad \widehat{\Gamma }=I_s\otimes \Theta +\tilde{\Gamma }\otimes \Theta \Omega , \\ \tilde{A}:=A(I_s-L)^{-1}, &{}\quad \tilde{b}^\top :=b^\top (I_s-L)^{-1}, &{}\quad \tilde{\Gamma }=(I_s-L)^{-1}-I_s:=(\tilde{\gamma }_{i,j}). \end{array} \end{aligned}$$

(10)

The corresponding order conditions for s-stage TASE$_r$ W-methods can then be obtained from the order conditions for an $(r\cdot s)$-stage W-method in Tables 1 and 2 with the coefficient matrices $\widehat{\tilde{A}}$, $\widehat{\tilde{b}}$ and $\widehat{\Gamma }$ of (10) and $\widehat{\tilde{c}}:=\widehat{\tilde{A}}\textbf{1}_{rs}=c\otimes \textbf{1}_r$. However, in [11] the authors have simplified the expression of the order conditions of TASE W-methods bringing all the calculations to the level of matrices of size s instead of $r\cdot s$. In doing so, the coefficients

$$\begin{aligned} \eta _k:=\textstyle \sum _{\ell =1}^r \omega _\ell \theta _\ell ^k,\quad k\ge 1, \end{aligned}$$

(11)

play a major role. Observe that the $\eta _k$ coefficients in (11) are just the coefficients appearing in the Neumann expansion of the TASE operator (7) in powers of $\tau W.$ In fact, general order conditions for order p are established in terms of the matrix coefficients $\tilde{A}$, $\tilde{b}$, $\tilde{\Gamma }$ and the coefficients $\{\eta _k\}_{k\ge 1}$ in [11, Theorem 2] (see also Sect. 5 below). The corresponding order conditions up to order $p=4$ are collected in Table 3 (see also [11, Table 4]), where the coefficient vector and matrix

(12)

are considered, with $\tilde{\Gamma }=(\tilde{\gamma }_{i,j})$ the strictly lower triangular matrix defined in (10).

Table 3 Order conditions for TASE W-methods $(\tilde{A}, \Gamma , \tilde{b})$, with $c=\tilde{A}\textbf{1}_s$ and $\rho$ and $\tilde{\Gamma }_{\mathbb {1}}$ given in (12), up to order 4 (plus the RK order conditions in Table 1)

From here, efficient TASE W-methods with $r=2$ were built in [11] with order of consistency $p=s=2,3,4$, small error coefficients and good linear stability properties such as L-stability (when $W=J$). The TASE W-methods so obtained were shown to be competitive, even via a sequential implementation (not exploiting the inherent parallelism), not only to the TASE-RK methods with $r=s$ in [2] and [5] but also to existing W-methods in the literature (for which $r=1$).

3 AMF-TASE W-Methods for Split ODEs

In many situations, the vector field f(y) of the ODE (1) is endowed with a natural splitting. This is, e.g., the case of the ODEs with the dimensional splitting arising after spatial discretization of parabolic PDEs by means of Finite Differences in d spatial dimensions. Let us now consider the split ODE

$$\begin{aligned} y'=f(y)=\textstyle \sum _{\nu =0}^d f_{\nu }(y) \end{aligned}$$

(13)

with $d\ge 1$ and the corresponding Jacobian splitting at $(t_n,y_n)$

$$\begin{aligned} J_n =\textstyle \sum _{\nu =0}^d J_{n,\nu }, \quad J_n := f'(y_n), \quad J_{n,\nu } :=f'_{\nu }(y_n),\;\; \nu =0,\ldots , d. \end{aligned}$$

(14)

The AMF [9, 20, 21] can be merged in TASE W-methods taking $W_\ell$ matrices from

$$\begin{aligned} I_M-\theta _\ell \tau W_\ell =\prod _{\nu =1}^d (I_M-\theta _\ell \tau J_{n,\nu }) \ = (I_M-\theta _\ell \tau J_{n,1}) (I_M-\theta _\ell \tau J_{n,2}) \cdots (I_M-\theta _\ell \tau J_{n,d}), \end{aligned}$$

(15)

in such a way that

$$\begin{aligned} \begin{aligned} W_\ell&=\sum _{\nu =1}^d J_{n,\nu } +(-\theta _\ell \tau )\sum _{1\le \nu _1<\nu _2\le d} J_{n,\nu _1}J_{n,\nu _2} + (-\theta _\ell \tau )^2\sum _{1\le \nu _1<\nu _2<\nu _3\le d} J_{n,\nu _1}J_{n,\nu _2}J_{n,\nu _3} \\&+ \cdots +(-\theta _\ell \tau )^{d-1}\prod _{\nu =1}^d J_{n,\nu } =J_n-J_{n,0}+\mathcal {O}(\tau ), \quad \tau \rightarrow 0. \end{aligned} \end{aligned}$$

(16)

Observe that we include in (13) a split term $f_{0}(y)$ whose Jacobian $J_{n,0}$ (14) is discarded from the AMF (15), i.e., we consider in general an inexact AMF. This allows to consider split terms that can be treated explicitly (e.g., a non stiff term) or split terms that can be troublesome to treat implicitly (e.g., split terms with non-structured Jacobians). Otherwise, when $f_{0}(y)=0$, the AMF approach will be regarded as exact AMF.

More generally, let us consider non-autonomous split ODE problems

$$\begin{aligned} u'(t)=F(t,u)=\textstyle \sum _{\nu =0}^d F_{\nu }(t,u), \quad u(t_0)=u_0, \quad t\in [t_0,t^*],\qquad u,F\in \mathbb {R}^M. \end{aligned}$$

(17)

The extension of the s-stage TASE$_r$ W-methods (6) with incorporated AMF approach (15) for the non-autonomous split ODE (17) reads

$$\begin{aligned} \left\{ \begin{aligned} K_i^{[\ell ,0]}:=K_i^{[0]}&= \tau F\Bigl (t_n+c_i\tau , u_n+\textstyle \sum _{j=1}^{i-1} a_{i,j} K_j\Bigr ) + \textstyle \sum _{j=1}^{i-1} \ell _{i,j} K_j,\\ \Bigl (I_M-\theta _\ell \tau \partial _u F_\nu (t_n,u_n)\Bigr ) K_i^{[\ell ,\nu ]}&= K_i^{[\ell ,\nu -1]} + \theta _\ell \rho _i \tau ^2 \partial _t F_\nu (t_n,u_n), \;\; \nu =1,\ldots ,d, \\ K_i^{[\ell ]}&=K_i^{[\ell ,d]},\;\; \ell =1,\ldots ,r,\\ K_i&=\textstyle \sum _{\ell =1}^r \omega _\ell K_i^{[\ell ]},\;\; i=1,\ldots ,s, \\ u_{n+1}&=u_n + \textstyle \sum _{i=1}^s b_i K_i, \end{aligned}\right. \end{aligned}$$

(18)

with $(c_i)_{i=1}^s= A\rho =c$ and $\rho =(\rho _i)_{i=1}^s= (I_s-L)^{-1}\textbf{1}_s.$ To prove this, just consider the augmented autonomous ODE in $\mathbb {R}^{1+M}$ as in (13) with $y:=(t,u^\top )^\top , \ f_0(y)= \bigl ( 1,F_0(t,u)^\top \bigr )^\top$, and $f_\nu (y)= \bigl ( 0,F_\nu (t,u)^\top \bigr )^\top ,\ \nu =1,\ldots , d$, whose split Jacobian is $f'(y_n)=\sum _{\nu =0}^d f'_\nu (y_n)$, then carrying out steps similar to those of [11, Subsection 3.2], taking $\mathcal {W}_\ell$ matrices according to (15)–(16).

Remark 1

For the particular situation with $d=1$ and $F_0(t,u)=0$ in (17), i.e., non-split ODEs, (18) is denoted as TASE ROW-method since it defines the extension of Rosenbrock-Wanner methods to the context of TASE operators and non-autonomous ODEs. Furthermore, observe that, under the same situation with $d=1$ and $F_0(t,u)=0$, approximating $\partial _u F(t_n,u_n)\approx W$ and $\partial _t F(t_n,u_n)\approx 0$ in (18) defines the class of TASE W-methods introduced in (6), in this case for non-autonomous ODEs (see also [11, Subsection 3.2]).

3.1 Linear Stability

We next describe the linear stability function of AMF-TASE W-methods (18), obtained when the method is applied to a linear test problem with splitting (19). In the context of (linear) PDE problems, this linear stability function plays a relevant role in the convergence of the numerical method to the exact solution of the PDE.

Theorem 1

The solution of an s-stage AMF-TASE$_r$ W-method (18) applied to the linear test problem with splitting

$$\begin{aligned} \begin{array}{c} y'=f(y)=\lambda y,\qquad \lambda =\lambda _0+\lambda _1+\cdots +\lambda _d, \end{array} \end{aligned}$$

(19)

is given by $y_{n+1}=R(z_0,z_1,\ldots ,z_d)y_n$, with

$$\begin{aligned} R(z_0,z_1,\ldots ,z_d)=1+(z\chi )b^\top \Bigl (I_s-\chi (z A+ L)\Bigr )^{-1}\textbf{1}_s, \end{aligned}$$

(20)

where $z_\nu =\tau \lambda _\nu$ $(\nu =0,\ldots ,d)$, $z=z_0+z_1+\cdots +z_d$,

$$\begin{aligned} \chi :=\textstyle \sum _{\ell =1}^r \dfrac{\omega _\ell }{\pi _\ell }\qquad \text {and}\qquad \pi _\ell :=\textstyle \prod _{\nu =1}^d (1-\theta _\ell z_\nu ),\quad \ell =1,\ldots ,r. \end{aligned}$$

(21)

Proof

By applying (18) to (19), it holds that

$$\begin{aligned} \begin{aligned} K_i^{[\ell ]}&=\frac{1}{(1-\theta _\ell z_d)} K_i^{[\ell ,d-1]}{=}\frac{1}{(1-\theta _\ell z_d)(1-\theta _\ell z_{d-1})} K_i^{[\ell ,d-2]}=\cdots {=}\frac{1}{\pi _\ell } K_i^{[\ell ,0]}{=}\frac{1}{\pi _\ell } K_i^{[0]}. \end{aligned} \end{aligned}$$

Thus, $K_i=\sum _{\ell =1}^r \frac{\omega _\ell }{\pi _\ell }K_i^{[0]}=\chi K_i^{[0]}$, $i=1,\ldots ,s$. From the first equation in (18) we get

$$\begin{aligned} K_i^{[0]}=z\Bigl (y_n+\chi \textstyle \sum _{j=1}^{i-1} a_{i,j} K_j^{[0]}\Bigr ) + \chi \textstyle \sum _{j=1}^{i-1} \ell _{i,j} K_j^{[0]},\qquad i=1,\ldots ,s. \end{aligned}$$

Solving for $K^{[0]}:=(K_1^{[0]},\ldots ,K_s^{[0]})^\top \in \mathbb {R}^s$ gives $K^{[0]}=\Bigl (I_s-\chi (z A+ L)\Bigr )^{-1}\textbf{1}_s\cdot (z y_n),$ which inserted in $y_{n+1}=y_n + \chi \sum _{i=1}^s b_i K_i^{[0]}$ concludes the proof. $\square$

Remark 2

Taking $d=1$ and $z_0=0$ in (20)–(21) gives the stability function of an s-stage TASE$_r$ W-method (6) for the linear test ODE $y'=\lambda y$ (without splitting). We also observe that, in terms of the matrices $\tilde{A}$, $\tilde{b}$ and $\tilde{\Gamma }$ of (10), the stability function (20) can alternatively be expressed as

$$\begin{aligned} R(z_0,z_1,\ldots ,z_d)=1+(z\chi )\tilde{b}^\top \Bigl (I_s-(z \chi ) \tilde{A} -(\chi -1)\tilde{\Gamma })\Bigr )^{-1}\textbf{1}_s. \end{aligned}$$

(22)

3.2 Main Result on Consistency

When $r=1$, the family of AMF-TASE$_1$ W-methods (18) coincides with the standard AMF W-methods introduced in [9, p. A2915, (4.7)–(4.8)]. In this particular case, these methods are again W-methods since every internal stage is related to the same W matrix of the form (16) with $\ell =1$. Hence, the order of consistency of an AMF W-method ($r=1$) is immediately obtained from the order of consistency of the underlying W-method. On the other hand, for $r\ge 2$, the methods (18) are no longer TASE W-methods as defined in (6) since now each stage $K_i^{[\ell ]}$ is related to the matrix $W_\ell$ in (16), which is different for each value of $\ell =1,\ldots ,r$. Therefore, deriving the consistency of the AMF-TASE W-method (18) from the consistency of the underlying TASE W-method requires a more elaborated analysis. In any case, Theorem 2 below, which is the main theoretical result of the manuscript, indicates that the s-stage AMF-TASE$_r$ W-method (18) preserves the order of consistency of the underlying s-stage TASE$_r$ W-method (6). Of course, the converse is trivially true due to the Remark 1.

Theorem 2

Let the s-stage TASE$_r$ W-method (6) be consistent of order $p\ge 1$. Then the associated s-stage AMF-TASE$_r$ W-method (18) is consistent of order p.

Proof

See Subsection 5.3.

Although the proof of Theorem 2 is postponed to Subsection 5.3 since it requires further technical details, to motivate the content of the following Sects. 4 and 5 we first illustrate in the Remark 3 below the main ideas of the proof.

Remark 3

In a first step, for the numerical solution of (1), we introduce a variant of the W-methods (2) (or (3)), where a different matrix $W_i$ of the form (16) is employed to compute each internal stage $K_i$, $i=1,\ldots ,s$. This will lead to additional order conditions on the method coefficients. The mentioned methods are thus of the form

$$\begin{aligned} \left\{ \begin{aligned} (I_M-\theta _i\tau W_i) K_i&= \tau f\Bigl (y_n+ \textstyle \sum _{j=1}^{i-1} a_{i,j} K_{j}\Bigr ) + \textstyle \sum _{j=1}^{i-1} \ell _{i,j} K_j, \quad i=1,\ldots ,s, \\ y_{n+1}&=y_n + \textstyle \sum _{i=1}^{s} b_i K_i, \end{aligned} \right. \end{aligned}$$

(23)

with matrices $W_i\in \mathbb {R}^{M\times M}$ which can be different from each other. Standard W-methods are obtained in the situation where $W_i=W$, $i=1,\ldots ,s$. The choice of considering these new W-methods derives from the fact that the AMF-TASE W-methods are a subclass of them. Indeed, as already noted in this section, the AMF-TASE W-methods are a variant of TASE W-methods (6) (in the case of autonomous problems, but this also applies to non-autonomous problems, see Remark 1) consisting of $W_\ell$ matrices (in place of W) defined in accordance with (16). Using this observation, we will study the consistency of the AMF-TASE W-methods (in Sect. 5) through the investigation of the consistency of the schemes (23) (in Sect. 4).

In a second step, we note that the order conditions of the AMF-TASE W-method (18) are automatically fulfilled, thanks to those of the corresponding TASE W-method.

Considering the approach above, the proof of Theorem 2 for arbitrary order p will be derived along the following steps.

• 1. General order conditions for the methods (23), with the matrices $W_i$ defined as in (16) taking $\ell =i$, will be obtained in Sect. 4.

• 2. The corresponding additional order conditions for the AMF-TASE W-methods (18) will be shown to hold in Sect. 5. For instance, to get order 4 we will see that each term in the sums appearing in Table 3 must be zero (see, e.g., (31) when $p=3$).

4 AMF W$_i$-methods

4.1 Formulation and Order Conditions of AMF W$_i$-methods up to Order $p=4$

Definition 1

An s-stage AMF W$_i$-method applied to (1) is given by the formula (23), where the matrices $W_i=W_i(\tau )$, $i=1,\ldots ,s$, have the particular form

$$\begin{aligned} W_i=\widehat{W}_1+(-\theta _i \tau ) \widehat{W}_2 + (-\theta _i \tau )^2 \widehat{W}_3 +\cdots + (-\theta _i \tau )^{d-1} \widehat{W}_d, \end{aligned}$$

(24)

with a fixed, but arbitrary, integer $d\ge 1$, and constant matrices $\widehat{W}_\nu$, $1\le \nu \le d$, which are independent of the method coefficients.

The motivation for the acronym of AMF W$_i$-methods and the selection of matrices $W_i$ of the form (24) comes from the AMF approach in (15)–(16).

Similarly as in Subsection 2.1, the method (23) can be expressed alternatively as

$$\begin{aligned} \left\{ \begin{aligned} \tilde{K}_i&= \tau f\Bigl (y_n+ \textstyle \sum _{j=1}^{i-1}\tilde{a}_{i,j} \tilde{K}_{j}\Bigr ) + \tau W_i \textstyle \sum _{j=1}^{i} \gamma _{i,j} \tilde{K}_j, \quad i=1,\ldots ,s, \\ y_{n+1}&=y_n + \textstyle \sum _{i=1}^{s} \tilde{b}_i \tilde{K}_i, \end{aligned} \right. \end{aligned}$$

(25)

where the coefficients $(A,L,\Theta ,b)$ and $(\tilde{A},\Gamma ,\tilde{b})$ satisfy the same relations as in (4)–(5), i.e. $\Gamma =\Theta +\Theta \tilde{\Gamma }$ and $\Theta =\text {Diag}(\theta _i)$. The formulation (25) will be helpful to derive the order conditions of AMF W$_i$-methods for consistency of arbitrary order p.

Since each internal stage $\tilde{K}_i$ in (25) is related to a different $W_i$ matrix, the consistency of these newly defined AMF W$_i$-methods needs additional order conditions to be fulfilled in comparison to those of standard W-methods (for which $W_i=W$, $\forall \; i=1,\ldots ,s$, i.e., each stage is associated to the same matrix W). Of course, when $\widehat{W}_2=\ldots =\widehat{W}_d=0$ in (24), we recover a W-method (3) and hence the order conditions for these latter methods form a subset of the order conditions for AMF W$_i$-methods.

The series expansion in terms of elementary differentials of f for the advancing solution of (25) will be obtained similarly as for W-methods in [17, p. 114–116]. For sake of clarity, we first illustrate the procedure for the derivation of the order conditions up to order $p=4$. We have that

$$\begin{aligned} \tilde{K}_i=\tau \cdot \tilde{K}_{i{|\tau =0}}^{(1)} + \tfrac{\tau ^2}{2} \cdot \tilde{K}_{i{|\tau =0}}^{(2)} + \tfrac{\tau ^3}{6} \cdot \tilde{K}_{i{|\tau =0}}^{(3)}+ \tfrac{\tau ^4}{24} \cdot \tilde{K}_{i{|\tau =0}}^{(4)}+O(\tau ^5), \end{aligned}$$

(26)

with $\tilde{K}_{i{|\tau =0}}^{(0)}:=\tilde{K}_{i{|\tau =0}}=0$ and $\tilde{K}_{i{|\tau =0}}^{(j)}$ denoting $\frac{d^j \tilde{K}_i}{d \tau ^j}(\tau =0)$. From (25) and using Leibniz’s rule [16, p. 144, (2.4)], we get for each $i=1,\ldots ,s$ that

$$\begin{aligned} \left\{ \begin{aligned} \tilde{K}_{i{|\tau =0}}^{(q)}&= q \bigl [f(u_i)\bigr ]^{(q-1)}_{|\tau =0} + q \Bigl [W_i \textstyle \sum _{j=1}^{i} \gamma _{i,j} \tilde{K}_j\Bigr ]^{(q-1)}_{|\tau =0}, \quad q\ge 1, \\ u_i&=y_n+ \textstyle \sum _{j=1}^{i-1}\tilde{a}_{i,j} \tilde{K}_{j}, \end{aligned} \right. \end{aligned}$$

(27)

where $\gamma _{i,j}:=\theta _i(\delta _{i,j}+\tilde{\gamma }_{i,j})$ -being $\delta _{i,j}$ the Kronecker delta- are the coefficients of the lower triangular matrix $\Gamma$ in (5) and the elementary differentials are evaluated at $y_n$.

By means of (24) and (27), we get

$$\begin{aligned} \begin{aligned} \tilde{K}_{i{|\tau =0}}^{(1)}&= f, \\ \tilde{K}_{i{|\tau =0}}^{(2)}&= 2 \Bigl (\textstyle \sum _{j} \tilde{a}_{i,j}\Bigr ) f'(f) + 2 \Bigl (\textstyle \sum _{j} \gamma _{i,j}\Bigr ) \widehat{W}_1 f, \\ \tilde{K}_{i{|\tau =0}}^{(3)}&= 3 \Bigl (\textstyle \sum _{j,k} \tilde{a}_{i,j} \tilde{a}_{i,k}\Bigr ) f''(f,f) + 6 \Bigl (\textstyle \sum _{j,k} \tilde{a}_{i,j} \tilde{a}_{j,k}\Bigr ) f'(f'(f)) + 6 \Bigl (\textstyle \sum _{j,k} \tilde{a}_{i,j} \gamma _{j,k} \Bigr ) f'(\widehat{W}_1 f) \\&\quad + 6 \Bigl (\textstyle \sum _{j,k} \gamma _{i,j} \tilde{a}_{j,k} \Bigr ) \widehat{W}_1 f'( f) + 6 \Bigl (\textstyle \sum _{j,k} \gamma _{i,j} \gamma _{j,k} \Bigr ) \widehat{W}_1^2 f + 6 (-\theta _i) \Bigl (\textstyle \sum _{j} \gamma _{i,j}\Bigr ) \widehat{W}_2 f,\\ \tilde{K}_{i{|\tau =0}}^{(4)}&= \tilde{K}_{i{|\tau =0}}^{(4,f,\widehat{W}_1)} + 24 \Bigl (\textstyle \sum _{j,k} \tilde{a}_{i,j} (-\theta _j) \gamma _{j,k} \Bigr ) f'(\widehat{W}_2 f) + 24 (-\theta _i) \Bigl (\textstyle \sum _{j,k} \gamma _{i,j} \tilde{a}_{j,k}\Bigr ) \widehat{W}_2 f'(f) \\ {}&\quad + 24 (-\theta _i) \Bigl (\textstyle \sum _{j,k} \gamma _{i,j} \gamma _{j,k}\Bigr ) \widehat{W}_2 \widehat{W}_1 f + 24 \Bigl (\textstyle \sum _{j,k} \gamma _{i,j} (-\theta _j) \gamma _{j,k}\Bigr ) \widehat{W}_1 \widehat{W}_2 f \\&\quad + 24 (-\theta _i)^2 \Bigl (\textstyle \sum _{j} \gamma _{i,j}\Bigr ) \widehat{W}_3 f , \end{aligned} \end{aligned}$$

(28)

where, for the sake of brevity, in $\tilde{K}_{i{|\tau =0}}^{(4,f,\widehat{W}_1)}$ we collect all the terms related to elementary differentials associated exclusively to f and $\widehat{W}_1$ (i.e., associated to W-methods). In (28) all the elementary differentials are evaluated at $y_n$. Observe that the first and second order derivatives of $\tilde{K}_i$ for $\tau =0$ coincide exactly with those of the W-methods (see [17, pp. 114–116]). Instead, $\tilde{K}^{(3)}_{i{|\tau =0}}$ contains a new elementary differential $\widehat{W}_2 f$, whereas $\tilde{K}^{(4)}_{i{|\tau =0}}$ contains five new elementary differentials. Of course, these elementary differentials need to be considered whenever $d\ge 3$ in (24) and $\widehat{W}_\nu \ne 0$, $\nu =2,3$.

Using the advancing solution of (25), along with (26) and (28), we can derive the order conditions of AMF W$_i$-methods (25)–(24) up to order 4.

Note that, considering (26)–(28) and (24), for the method (25) to reach order $p=1$ or $p=2$ no additional order conditions are needed apart from the ones for W-methods in Table 2. However, for order $p=3$, the method needs to satisfy $\sum _{i,j} \tilde{b}_i\gamma _{i,j} \widehat{W}_1 f=\mathcal {O}(\tau ^2)$ (from (26) and $\tilde{K}_{i{|\tau =0}}^{(2)}$ in (28)), which implies the additional order condition

$$\begin{aligned} \tilde{b}^\top (-\Theta )\Gamma \textbf{1}_s=0. \end{aligned}$$

(29)

Furthermore, observe that considering (10) and (9), this new condition (29) for order $p=3$ applied to the AMF-TASE W-method (18) reads

$$\begin{aligned} \widehat{\tilde{b}}^\top (I_s\otimes \Theta )\widehat{\Gamma }\textbf{1}_{rs}{=}(\tilde{b}^\top \textbf{1}_{s})\cdot (\omega ^\top \Theta ^2\textbf{1}_{r}){+}(\tilde{b}^\top \tilde{\Gamma }\textbf{1}_{s})\cdot (\omega ^\top \Theta ^2\Omega \textbf{1}_{r}){=}\eta _2\cdot \tilde{b}^\top (\tilde{\Gamma }+I_s) \textbf{1}_{s}{=}\eta _2\cdot \tilde{b}^\top \rho {=}0. \end{aligned}$$

(30)

However, (30) is automatically fulfilled, since considering all the order three conditions of TASE W-methods in Table 3 it is not difficult to check that

$$\begin{aligned} \eta _2\cdot \tilde{b}^\top \rho + \eta _1^2\cdot \tilde{b}^\top \tilde{\Gamma }\rho =0 \iff \eta _2\cdot \tilde{b}^\top \rho =\eta _1^2\cdot \tilde{b}^\top \tilde{\Gamma }\rho =0, \end{aligned}$$

(31)

and hence order $p=3$ holds for the AMF-TASE W-method (18).

Similar calculations allow to derive the order conditions of AMF W$_i$-methods also for order $p=4$. Table 4 contains the new order conditions of AMF W$_i$-methods up to order $p=4$.

Table 4 Order conditions for the AMF W$_i$-methods $(\tilde{A}, \Gamma , \tilde{b})$ in (25)–(24), with $\Theta =Diag(\theta _i)$, $c=\tilde{A}\textbf{1}_s$, up to order 4 (plus the order conditions for RK and W-methods in Tables 1 and 2).

4.2 General Order Conditions for AMF W$_i$-methods

With the aim of deriving general order conditions for AMF W$_i$-methods in (25)–(24), new types of graphs (Butcher trees) for the elementary differentials related to the matrices $\widehat{W}_\nu$, $\nu \ge 2$, need to be defined. In particular, in the following we extend the set of $\mathbb{T}\mathbb{W}$-trees associated with a W-method [17, pp. 114–116]. We denote with $\mathbb {T}$ the set of rooted trees [16, pp. 145–148] with black vertices (or meager vertices) associated to f and its elementary differentials, as considered for RK and Rosenbrock methods. A tree $\mathcal {T} \in \mathbb {T}$ is denoted by $\mathcal {T}=[\mathcal {T}_1,\ldots ,\mathcal {T}_q]_\bullet$, with $\mathcal {T}_0=\bullet$. The set $\mathbb{T}\mathbb{W}$ is obtained by adding trees with at least one white vertex (or fat vertex), associated to the matrix W (here $W=\widehat{W}_1$), i.e., $\mathbb{T}\mathbb{W}$ is composed of rooted trees whose end vertices are black and the white vertices can only be singly-branched.

Definition 2

(Set of $\mathbb {T\widehat{W}}$-trees). The set of $\mathbb {T\widehat{W}}$-trees is defined as follows: it includes the set of $\mathbb {T}$-trees and also the trees obtained by connecting a white $\nu$-vertex, i.e., a white vertex of weight $\nu \ge 1$ ($\nu \in \mathbb {N}$), to a tree $\mathcal {T}_1$. A graph of this type is denoted by when $\nu \ge 2$, and by when $\nu =1$. Given an integer $d\ge 1$, the set $\mathbb {T\widehat{W}}^{(d)}$ is the subset of $\mathbb {T\widehat{W}}$ with trees whose $\nu$-vertices fulfil $1\le \nu \le d$.

Observe that $\mathbb{T}\mathbb{W} \subseteq \mathbb {T\widehat{W}}$ and that $\mathbb {T\widehat{W}}$ is composed of rooted trees whose end vertices are black and white $\nu$-vertices can only be singly branched. The following definition extends the concepts of order, density and cardinality for $\mathbb {T}$-trees (see, e.g., [3, pp. 153-155]) to $\mathbb {T\widehat{W}}$-trees. Observe that for a tree $\mathcal {T}\in \mathbb{T}\mathbb{W}$, such values coincide with the corresponding values for the associated tree in $\mathbb {T}$ with all its vertices of black type (see Fig. 1).

Definition 3

(Order, density and cardinality of $\mathbb {T\widehat{W}}$-trees). Given a tree $\mathcal {T}\in \mathbb {T\widehat{W}}$, let us consider the associated tree $\overline{\mathcal {T}} \in \mathbb{T}\mathbb{W}$ obtained by substituting each $\nu$-vertex of the form in the tree $\mathcal {T}$ with a sequence of $\nu$ consecutive white vertices. Then, the order, density and cardinality of $\mathcal {T}$ are $\rho (\mathcal {T})=\rho (\overline{\mathcal {T}})$, $\gamma (\mathcal {T})=\gamma (\overline{\mathcal {T}})$ and $\alpha (\mathcal {T})=\alpha (\overline{\mathcal {T}})$, respectively.

Figure 1 shows how to obtain the associated tree $\overline{\mathcal {T}} \in \mathbb{T}\mathbb{W}$ from a tree $\mathcal {T}\in \mathbb {T\widehat{W}}$.

Fig. 1

Associated tree $\overline{\mathcal {T}} \in \mathbb{T}\mathbb{W}$ (and $\widetilde{\mathcal {T}} \in \mathbb {T}$) from a tree $\mathcal {T}\in \mathbb {T}\widehat{\mathbb {W}}$ (see Definition 3)

From Definition 3 it is not difficult to check that for $\mathcal {T}\in \mathbb {T\widehat{W}}$ it holds

(32)

(33)

while $\alpha (\mathcal {T})$ counts the number of possible different monotonic labellings of the associated tree $\overline{\mathcal {T}} \in \mathbb{T}\mathbb{W}$ (or $\widetilde{\mathcal {T}} \in \mathbb {T}$, see [16, Definition 2.5, p. 147]). We also observe that the order of a tree $\mathcal {T} \in \mathbb {T\widehat{W}}$ is the number of black vertices plus the number of white $\nu$-vertices, each weighted with its weight $\nu$.

Definition 4

(Elementary differentials of $\mathbb {T\widehat{W}}$-trees). Given the ODE (1) and a tree $\mathcal {T} \in \mathbb {T\widehat{W}}$, the elementary differential $F(\mathcal {T})$ is a function $F(\mathcal {T}):\mathbb {R}^M \rightarrow \mathbb {R}^M$ defined recursively as $F(\mathcal {T}_0)(y)=f(y)$ and

where $\widehat{W}_\nu$, $\nu \ge 1$, are the constant matrices in (24).

Definition 5

Given the s-stage AMF W$_i$-method (25)–(24), for $\mathcal {T} \in \mathbb {T\widehat{W}}$ the function $\Phi (\mathcal {T})$ is defined recursively as $\Phi (\mathcal {T}_0)=\mathbf {1_s}$ and

where $\varvec{*}$ is the componentwise product of vectors in $\mathbb {R}^s$.

Fig. 2

$\Phi (\mathcal {T})$ in (35) and elementary differential $F(\mathcal {T})(y)$ in (34) for a $\mathbb {T\widehat{W}}$-tree of type

Figure 2 shows an example on the computation of the function $\Phi (\mathcal {T})$. The following Theorem 3 states the order conditions of arbitrary order p for AMF W$_i$-methods.

Theorem 3

The s-stage AMF W$_i$-method $({\tilde{A}},{\Gamma },{\tilde{b}})$ in (25), with $W_i$ given in (24) for $d\ge 1$, is consistent of order $p\ge 1$ if and only if

$$\begin{aligned} {\tilde{b}}^\top \Phi (\mathcal {T})=\left\{ \begin{array}{ll} 1/\gamma (\mathcal {T}), &{} \text {if}\quad \mathcal {T}\in \mathbb {T},\quad \rho (\mathcal {T})\le p,\\ 0, &{} \text {if}\quad \mathcal {T}\in \mathbb {T\widehat{W}}^{(d)}\setminus \mathbb {T},\quad \rho (\mathcal {T})\le p, \end{array} \right. \end{aligned}$$

(36)

where the functions on trees $\rho (\mathcal {T})$ and $\gamma (\mathcal {T})$ are defined in Definition 3 and $\Phi (\mathcal {T})$ in Definition 5. Recall that d is the number of terms in the splitting (24), whereas the set of trees $\mathbb {T\widehat{W}}^{(d)}$ has been introduced in Definition 2.

Proof

See Subsection 7.1.

Table 5 below collects the trees in $\mathbb {T\widehat{W}} \setminus \mathbb{T}\mathbb{W}$ up to order 4 with related elementary differentials. These trees and elementary differentials are associated to the additional order conditions of AMF W$_i$-methods with respect to the classical W-methods in Table 4 (respectively, in consecutive order).

Table 5 Representation of trees in $\mathbb {T\widehat{W}} {\setminus } \mathbb{T}\mathbb{W}$ up to order 4 and related elementary differentials, cardinality, density and function $\Phi (\mathcal {T})$; $c=\tilde{A}\textbf{1}_s$

5 Consistency of AMF-TASE W-methods

This section is devoted to prove the Theorem 2 on the consistency of the AMF-TASE W-method (18). Since this latter method is a particular AMF W$_i$-methods with $r \cdot s$ stages $K_i^{[\ell ]}$ ($i=1,\ldots ,s$, $\ell =1,\ldots ,r$) as defined in (23) (see also (25)) with the matrices $W_i$ defined as in (16) taking $\ell =i$, its order conditions are just obtained from Theorem 3 with the corresponding coefficient matrices and vector $(\widehat{\tilde{A}},{\widehat{\Gamma }},\widehat{\tilde{b}})$ of dimension $r\cdot s$ in (10) and (8). It will be shown that the corresponding order conditions for order p are automatically satisfied from the order p conditions of the underlying TASE W-method (6). To see this, we will first recall in Subsection 5.1 the expression in [11, Theorem 2] for the order conditions of TASE W-methods in terms of coefficients matrices and vectors of dimension s (instead of $r\cdot s$) and we prove that they are fulfilled addend by addend. This will be crucial to prove that the order conditions corresponding to trees $\mathcal {T}\in \mathbb {T\widehat{W}}{\setminus } \mathbb {T}$ for the AMF-TASE W-method are automatically satisfied (also addend by addend) in Subsection 5.2.

5.1 Consistency of TASE W-methods

General conditions for order p on the coefficients of a TASE W-method (6) were formulated in [11, Theorem 2] in terms of $\{\eta _k\}_{k=1}^{p-1}$ in (11) and $(\tilde{A},\tilde{\Gamma },\tilde{b})$ in (10) as Theorem 4 below indicates.

Theorem 4

[11, Theorem 2]. The s-stage TASE$_r$ W-method (6) has order p if and only if

$$\begin{aligned} \tilde{b}^\top \phi (\mathcal {T})=\left\{ \begin{array}{ll} 1/\gamma (\mathcal {T}), &{} \text {for}\quad \mathcal {T}\in \mathbb {T},\quad \rho (\mathcal {T})\le p,\\ 0, &{} \text {for}\quad \mathcal {T}\in \mathbb{T}\mathbb{W}\setminus \mathbb {T},\quad \rho (\mathcal {T})\le p, \end{array} \right. \end{aligned}$$

(37)

where the function $\phi (\mathcal {T})\in \mathbb {R}^s$ is defined as follows

with parameters $\{\eta _k\}_{k\ge 1}$ defined in (11) and $\tilde{\Gamma }_{\mathbb {1}}=\tilde{\Gamma }+I_s$ defined in (12).

Figure 3 illustrates the computation of order conditions for TASE W-methods at the level of matrices and vectors of dimension s.

Fig. 3

Graphical interpretation of the computation of order conditions for TASE W-methods at the level of matrices and vectors of dimension s in Theorem 4 taking into account all the possible partitions of the natural number $k=3$

As observed in (31) at the end of Sect. 2, it is not difficult to check from Table 3 that up to order 4 the conditions of TASE W-methods are equivalent to annihilating each addend of $\tilde{b}^\top \phi (\mathcal {T})$, with $\phi (\mathcal {T})$ in (38), whenever $\mathcal {T}\in \mathbb{T}\mathbb{W}{\setminus } \mathbb {T}$. Theorem 5 together with Remark 4 below indicate that this property also holds for arbitrary order p.

Theorem 5

Let an s-stage TASE$_r$ W-method (6) have order $p\ge 2$. For all integers $k\ge 1$ and trees $\widehat{\mathcal {T}}=\mathcal {T}_0$ or $\widehat{\mathcal {T}}=[\mathcal {T}_1,\ldots ,\mathcal {T}_q]_\bullet \in \mathbb{T}\mathbb{W}$ ($q\ge 1$) with $k+\rho (\widehat{\mathcal {T}})\le p$, it holds that

$$\begin{aligned} \displaystyle \sum _{j=1}^k \Bigl (\displaystyle \sum _{\begin{array}{c} 1\le n_1,\ldots ,n_j\le k \\ n_1+\cdots +n_j=k \end{array} } \eta _{n_1}\cdots \eta _{n_j}\Bigr )\ \tilde{b}^\top \tilde{\Gamma }^{j-1} \tilde{\Gamma }_{\mathbb {1}} \phi (\widehat{\mathcal {T}})=0 \Longrightarrow \eta _{n_1}\cdots \eta _{n_j} \tilde{b}^\top \tilde{\Gamma }^{j-1} \tilde{\Gamma }_{\mathbb {1}}\phi (\widehat{\mathcal {T}})=0, \end{aligned}$$

(39)

for all integers $1\le n_1,\ldots ,n_j\le k$ with $n_1+\cdots +n_j=k$ ($j=1,\ldots ,k$), where the function $\phi (\mathcal {T})\in \mathbb {R}^s$ is given by (38).

Proof

See Subsection 7.2.

Remark 4

Theorem 5 implies that in the order conditions (37) of TASE$_r$ W-methods for trees $\mathcal {T}\in \mathbb{T}\mathbb{W}{\setminus } \mathbb {T}$, $\rho (\mathcal {T})\le p$, each addend taken individually is zero in the situations of the third and fourth statement of (38). This observation also applies to the second statement, as explained in Remark 6 in Subsection 7.2, after the proof of Theorem 5.

5.2 Case of AMF-TASE W-methods

As indicated in Sect. 3 and Subsection 4.1, an s-stage AMF-TASE$_r$ W-method in the formulation (18) is a particular $(r\cdot s)$-stage AMF W$_i$-method (23) applied to a non-autonomous ODE. Hence, its order conditions can be thus derived using Theorem 3 with the coefficient matrices $(\widehat{\tilde{A}},\widehat{\Gamma },\widehat{\tilde{b}})$ of size $r\cdot s$ in (10) and (8). Similarly as in Subsection 5.1, the following Theorem 6 allows to simplify the order conditions of the AMF-TASE W-method by bringing all calculations to the level of coefficient matrices and vectors of dimension s (instead of $r\cdot s$). This theorem, along with the Theorems 4 and 5 and the Remark 4, will be crucial to prove that the AMF-TASE W-method retains the order of consistency of its underlying TASE W-method (6).

Theorem 6

Let us consider an s-stage AMF-TASE$_r$ W-method (18) with coefficients $(\widehat{\tilde{A}}, \widehat{\Gamma },\widehat{\tilde{b}})$ in (10) and (8), and the parameters $\{\eta _k\}$ in (11). The corresponding function $\Phi (\mathcal {T})\in \mathbb {R}^{rs}$ in (35) fulfils for all $\mathcal {T}\in \mathbb {T\widehat{W}}$

$$\begin{aligned} (I_s\otimes \Omega )\Phi (\mathcal {T})&=\varphi (\mathcal {T})\otimes \textbf{1}_r, \end{aligned}$$

(40)

$$\begin{aligned} \widehat{\tilde{A}}\Phi (\mathcal {T})&=\tilde{A}\varphi (\mathcal {T})\otimes \textbf{1}_r, \end{aligned}$$

(41)

$$\begin{aligned} \widehat{\tilde{b}}^\top \Phi (\mathcal {T})&=\tilde{b}^\top \varphi (\mathcal {T}), \end{aligned}$$

(42)

where $\varphi (\mathcal {T})\in \mathbb {R}^s$ satisfies

where

$$\begin{aligned} \begin{array}{c} p_m :=\displaystyle \sum _{q=l_0+\cdots +l_{m-1}}^{l_1+\cdots +l_{m}} \nu _q \quad (\text {with } l_0:=1 \text { and } m=1,\ldots ,i)\\ \text {fulfil} \quad p_1+\cdots +p_i=\nu _1+\cdots +\nu _j=k. \end{array} \end{aligned}$$

(44)

The s-stage AMF-TASE$_r$ W-method (18) with coefficients $(\widehat{\tilde{A}}, \widehat{\Gamma },\widehat{\tilde{b}})$ in (10) and (8) reaches order p if and only if

$$\begin{aligned} \tilde{b}^\top \varphi (\mathcal {T})=\left\{ \begin{array}{ll} 1/\gamma (\mathcal {T}), &{} \text {if}\quad \mathcal {T}\in \mathbb {T},\quad \rho (\mathcal {T})\le p,\\ 0, &{} \text {if}\quad \mathcal {T}\in \mathbb {T\widehat{W}}^{(d)} \setminus \mathbb {T},\quad \rho (\mathcal {T})\le p, \end{array} \right. \nonumber \\ \end{aligned}$$

(45)

with $d\ge 1$ defined by the splitting (17).

Proof

See Subsection 7.3.

Remark 5

From Theorem 4, let us observe that for trees $\mathcal {T}\in \mathbb{T}\mathbb{W}$ it holds that $\varphi (\mathcal {T})=\phi (\mathcal {T})$, with $\phi (\mathcal {T})$ given in (38) corresponding to the underlying TASE W-method. This can also be observed from (43) taking $\nu _1=\ldots =\nu _k=1$, with $j=k$, which inserted in (44) gives $p_m=l_m$, $m=1,\ldots ,k$, and therefore the same expression as in (38) is obtained.

In the spirit of Figs. 3 and 4 illustrates with an example the computation of order conditions (45) for AMF-TASE W-methods at the level of matrices and vectors of dimension s.

Fig. 4

Graphical interpretation of the computation of order conditions for AMF-TASE W-methods at the level of matrices and vectors of dimension s in Theorem 6 with $\nu _1=5$, $\nu _2=1$ and $\nu _3=2$ ($k=8$ and $j=3$)

Table 6 contains the additional order conditions up to order four to be fulfilled by the s-stage AMF-TASE$_r$ W-method (18) corresponding to trees $\mathcal {T}\in \mathbb {T\widehat{W}}^{(d)} \setminus \mathbb{T}\mathbb{W}$ (with $d\ge 3$). They correspond to those in Table 5 but computed with the coefficient matrices and vectors $(\tilde{A}, \tilde{\Gamma },\tilde{b})$ of dimension s given by (4)-(5) and the parameters $\{\eta _k\}_{k=1}^3$ in (11). It is observed that each addend in the new order conditions $\tilde{b}^\top \varphi (\mathcal {T})=0$ already appears as an addend in the original order conditions $\tilde{b}^\top \phi (\mathcal {T})=0$ for TASE W-methods (6) in Table 3. According to Theorem 5 and Remark 4, these new additional order conditions are also automatically satisfied addend by addend.

Table 6 Additional order conditions of orders $p=3$ and $p=4$ for s-stage AMF-TASE$_r$ W-methods (18) corresponding to trees and elementary differentials in Table 5 computed with the coefficient matrices and vectors $(\tilde{A}, \tilde{\Gamma },\tilde{b})$ of dimension s given by (4)–(5), $c=\tilde{A}\textbf{1}_s$, the matrix $\tilde{\Gamma }_{\mathbb {1}}$ in (12) and the parameters $\{\eta _k\}_{k=1}^3$ in (11)

5.3 Proof of Theorem 2

We are finally in the position of giving a proof for the main result of the paper on the consistency of AMF-TASE W-methods (Theorem 2).

Proof of Theorem 2

The order conditions for trees $\mathcal {T}\in \mathbb{T}\mathbb{W}$, with $\rho (\mathcal {T})\le p$, are automatically satisfied taking into account the order conditions of the underlying TASE W-method. Indeed, from Remark 5, we have $\tilde{b}^\top \varphi (\mathcal {T})=\tilde{b}^\top \phi (\mathcal {T})$ for the functions $\varphi (\mathcal {T})$ in (43) and $\phi (\mathcal {T})$ in (38), and therefore such order conditions are satisfied according to (37).

For a tree $\mathcal {T}\in \mathbb {T\widehat{W}}^{(d)} {\setminus } \mathbb{T}\mathbb{W}$, we consider the associated tree $\overline{\mathcal {T}} \in \mathbb{T}\mathbb{W}$, with $\rho (\mathcal {T})=\rho (\overline{\mathcal {T}})$, obtained by substituting $\nu$-vertices of the form in the tree $\mathcal {T}$ with a sequence of $\nu$ consecutive white vertices (see Definition 3 and Fig. 1). Due to (44), each addend in the expression of $\tilde{b}^\top \varphi (\mathcal {T})$ obtained from (43) for the tree $\mathcal {T}$ appears as an addend in the expression of $\tilde{b}^\top \phi (\overline{T})$ obtained from (38) for the tree $\overline{\mathcal {T}}$. Hence, considering Theorem 5 and Remark 4, each addend appearing in the expression of $\tilde{b}^\top \varphi (\mathcal {T})$ must be zero. $\square$

6 Numerical Illustration

This section is devoted to illustrate numerically that the order of consistency provided by AMF-TASE W-methods (18) coincides with that of the corresponding underlying TASE W-method (6) as stated in Theorem 2. The goal of this section also lies in showing that the AMF-TASE W-methods proposed here are advantageous to and/or competitive with well-known good performing AMF W-methods from the literature, when the AMF approach is either exact (Subsections 6.1 and 6.3) or inexact (Subsection 6.2). From [11, Subsection 6.1], we consider the following time integrators endowed with AMF. On the one hand,

• W3b (resp., ROW3b): the L-stable W-method with $s=3$ and $p=2$ (resp., the Rosenbrock-Wanner method with $p=3$ when $W=J+\mathcal {O}(\tau )$) given in [13, p. 573];

• TASE$_3$-RK3 (BFM): the TASE-RK method with $r=3$, $s=p=3$ derived in [2];

• TASE$_3$-RK3 (CMR): the TASE-RK method with $r=3$, $s=p=3$ derived in [5];

• TASE$_2$-W3: the L-stable TASE W-method with $r=2$, $s=p=3$ derived in [11, Subsection 5.2].

And on the other hand,

• W-Kutta3/8 (resp., ROW-Kutta3/8): the W-method, based on the Kutta’s 3/8-rule method, with $s=4$, $p=3$ and $\theta =\tfrac{1}{2}$ (resp., the Rosenbrock-Wanner method with $p=4$ when $W=J+\mathcal {O}(\tau )$) given in [12, p. 154];

• TASE$_4$-RK4 (BFM): the TASE-RK method with $r=4$, $s=p=4$ derived in [2];

• TASE$_4$-RK4 (CMR): the TASE-RK method with $r=4$, $s=p=4$ derived in [5];

• TASE$_2$-W4: the L-stable TASE W-method with $r=2$, $s=p=4$ derived in [11, Subsection 5.2].

6.1 A 3D Nonlinear PDE from Combustion Theory

We consider the 3D nonlinear reaction-diffusion problem from combustion theory [24]:

$$\begin{aligned} c_t =\Delta c -D\,c\, e^{-\delta /T}, \;\; L\, T_t =\Delta T+ \alpha D\, c\, e^{-\delta /T},\;\; t\in [0,0.2], \;\; (x,y,z)\in (0,1)^3, \end{aligned}$$

(46)

with initial conditions $c(x,y,z,0)=T(x,y,z,0)=1$, homogeneous Neumann boundary conditions at $x=0,\,y=0,\,z=0$ and Dirichlet conditions $c(x,y,z,t)=T(x,y,z,t)=1$ for $x=1,\,y=1,\,z=1$. The parameters in this case are $L=0.9$, $\alpha =1$, $\delta =20$, $D=R e^{\delta }/(\alpha \delta )$ and $R=5$. The solution of this PDE presents a reaction front which propagates towards the boundary planes $x=y=z=1$ where a boundary layer appears and the front reaches the boundary at a time close to $t=0.3$. To numerically solve this problem with fixed stepsizes we have selected a final time $t^*=0.2$.

We consider a spatial discretization based on second order central differences on the uniform mesh $(x_i,y_j,z_k)=\left( (i-\frac{1}{2})h, (j-\frac{1}{2})h, (k-\frac{1}{2})h \right)$, $i,j,k=1,\ldots ,N$, with $h=\left( N+\frac{1}{2}\right) ^{-1}$ and external points $x_0=y_0=z_0=-h/2$ for the discretization of the Neumann boundary conditions. Therefore, we obtain a semidiscrete ODE system $U'=f(U)=f_R(U)+f_x(U)+f_y(U)+f_z(U)$ of dimension $M=2N^3$, where $f_R$ contains the discretization of the reaction term, whereas $f_x,f_y,f_z$ contain the discretization of the diffusion terms in each spatial variable, respectively. Here $U=(C_{ijk},T_{ijk})_{i,j,k=1}^N$ stores approximated values $C_{ijk}=c(x_i,y_j,z_k,t)$ and $T_{ijk}=T(x_i,y_j,z_k,t)$, $i,j,k=1,\ldots ,N,$ for c and T at the grid points. We have considered a corresponding four-term splitting of the Jacobian matrix $J=f'(U_n)$, given by $J=J_R+J_x+J_y+J_z$, where $J_R=f'_R(U_n)$, $J_x=f'_x(U)$, $J_y=f'_y(U_n)$ and $J_z=f'_z(U_n)$. The AMF-TASE W-methods will be applied to the semi-discretization of (46) with fixed stepsize $\tau =0.2\cdot 2^{-j}$, $1\le j\le 16$, with $J_R$ included in the AMF (15). Observe that, at each integration step, the linear systems of type $(I-\theta _\ell \tau J_R)K=V$ are reduced to solve $N^3$ systems of dimension 2, whereas the linear systems associated to the diffusion part involving the matrices $J_x,J_y,J_z$ are reduced to solve $6 N^2$ tridiagonal systems of dimension N. Here, we consider the values $N=40$ and $N=80$. For each grid, a time-accurate reference ODE solution was previously computed with the integrator DOP853 [15, 16] and error tolerance $TOL=10^{-15}$.

In Fig. 5, it is observed that each method performs with the expected temporal order of convergence $p=3$ (plots on the top) and $p=4$ (plots on the bottom), respectively, both when $N=40$ (plots on the left) and $N=80$ (plots on the right). Here we have skipped the data corresponding to errors above $10^0$ and also those obtained after attaining the precision of the ODE reference solution. Observe that in this first PDE problem, the AMF approach is exact, and the methods AMF-ROW3b and AMF-ROW-Kutta3/8 are Rosenbrock-Wanner methods applied with $W=J+\mathcal {O}(\tau )$, which leads to the corresponding orders $p=3$ and $p=4$. Furthermore, it is observed that, for smaller stepsizes, the methods AMF-TASE$_2$-W3($\theta _1=0.5,\theta _2=0.7$) and AMF-TASE$_2$-W4($\theta _1=0.5,\theta _2=0.7$) provide errors about $10^{-2}$ times smaller than the corresponding errors given by AMF-TASE$_3$-RK3 (CMR) and AMF-TASE$_3$-RK3 (BFM), and AMF-TASE$_4$-RK4 (CMR) and AMF-TASE$_4$-RK4 (BFM), respectively. This is also in agreement with the size of the corresponding error coefficients for these methods as presented in [11, Tables 5-6].

Fig. 5

Stepsize vs Error in the $\ell _\infty$-norm for the time integration of the semi-discrete 3D combustion model (46) with the methods AMF-TASE$_3$-RK3 (CMR), AMF-TASE$_3$-RK3 (BFM), AMF-ROW3b and AMF-TASE$_2$-W3($\theta _1=0.5,\theta _2=0.7$) -plots on the top-, and AMF-TASE$_4$-RK4 (CMR), AMF-TASE$_4$ -RK4 (BFM), AMF-ROW-Kutta3/8 and AMF-TASE$_2$ -W4($\theta _1=0.5,\theta _2=0.7$) -plots on the bottom- when $N=40$ (plots on the left) and $N=80$ (plots on the right). Dashed straight lines with slope three (plots on the top) and four (plots on the bottom) are included to compare the temporal orders of convergence

6.2 A 3D Linear Parabolic PDE with Mixed Derivatives

We next consider the linear advection–diffusion PDE with constant coefficients and mixed derivatives in three spatial dimensions

$$\begin{aligned} \partial _t u = \textstyle \sum _{i=1}^3 d_{ii} \,\partial ^2_{x_i x_i} u + 2\sum _{1\le i<j\le 3} d_{ij} \,\partial ^2_{x_i x_j} u + \sum _{i=1}^3 a_{i} \,\partial _{x_i} u + g(t,\textbf{x}) \end{aligned}$$

(47)

for $\textbf{x}=(x_1,x_2,x_3)^\top \in (0,1)^3$, $t\in [0,1]$, where $g(t,\textbf{x})$ is selected such that the exact solution of (47) is (with $\beta =1$)

$$\begin{aligned} u(t,\textbf{x})= \textrm{e}^{\beta t} \bigg ( 4^3 \textstyle \prod _{j=1}^3 x_j(1-x_j) + \sum _{j=1}^3 \Bigl (x_j+\frac{1}{j+2}\Bigr )^2\bigg ). \end{aligned}$$

(48)

An initial condition and non-homogeneous time-dependent Dirichlet boundary conditions are imposed according to the exact solution (48). Furthermore, we take $d_{ii}=1$, $i=1,2,3$, $d_{ij}=0.5$, for $i\ne j$, and $a_1=-2$, $a_2=1$, $a_3=-0.5$. We must note that this kind of problems are tightly connected with PDE models in finance, such as the Heston model [18, 19], the Heston-Hull-White model [14] or some SABR/LIBOR market models [22]. We apply the MOL approach on a uniform grid with mesh-width $\Delta x_i=1/(N+1)$, $1\le i\le 3$, with $N=40$ and $N=80$, considering standard second order central differences in each spatial direction for the first and second order spatial derivatives (observe that there are no spatial errors since the PDE solution is a polynomial of second degree in each spatial variable). The following semi-discretized system of dimension $N^3$ is obtained

$$\begin{aligned} U' = J U+ G(t),\qquad J=J_0+J_1+J_2+J_3, \end{aligned}$$

(49)

where the matrix $J_i$ corresponds to the discretization of $d_{ii} \,\partial ^2_{x_i x_i} u+a_{i} \,\partial _{x_i} u$, $i=1,2,3$, and $J_0$ is associated to the discretization of the mixed derivatives terms $2\sum _{1\le i<j\le 3} d_{ij} \,\partial ^2_{x_i x_j} u$. G(t) stores the discretization of the term $g(t,\textbf{x})$ and also the terms due to non-homogeneous boundary conditions. AMF-TASE W-methods will be applied to (49) with fixed stepsize $\tau =2^{-j}$, $2\le j\le 17$, and $J_0$ discarded from the AMF (15) since its inclusion in the AMF would make the matrices appearing in the linear systems to be strongly coupled, also destroying the sparse bounded structure of the matrices involving pure derivatives. Consequently, the inclusion of $J_0$ in the AMF would increase considerably the LU store requirement and the cost of the LU linear system solutions [14, 19]. Therefore, in this example we consider an inexact AMF. Observe that the AMF approach avoids solving linear systems $(I-\theta _\ell \tau W)K=V$ of dimension $N^3$ associated to the TASE W-method (6), with a given approximation W of the exact Jacobian J in (49), and instead simplifies the linear algebra to solving the linear systems $(I-\theta _\ell \tau J_i)K=V$ ($i=1,2,3$) associated to the pure diffusion part which are reduced to solve $3 N^2$ tridiagonal systems of dimension N.

In Fig. 6, it is observed that each method performs with the expected temporal order of convergence $p=3$ (plots on the top) and $p=4$ (plots on the bottom), both when $N=40$ (plots on the left) and $N=80$ (plots on the right), with the exception of the methods AMF-W3b and AMF-W-Kutta3/8 for which the displayed order is one unit less. This is in agreement with the fact that the W-methods W3b and W-Kutta3/8 attain only order $p=2$ and $p=3$, respectively, when the matrix W does not fulfil $W=J+\mathcal {O}(\tau )$, which is the case in the present situation of inexact AMF due to the inclusion of mixed derivatives terms in the elliptic operator of the PDE. On the other hand, as in the previous numerical example, for the smaller stepsizes the relative size of the displayed errors of the methods is in agreement with the size of the corresponding error coefficients.

Fig. 6

Stepsize vs Error in the $\ell _\infty$-norm for the time integration of the semi-discrete 3D linear model (47)–(48) with the methods AMF-TASE$_3$ -RK3 (CMR), AMF-TASE$_3$ -RK3 (BFM), AMF-ROW3b and AMF-TASE$_2$ -W3($\theta _1=0.5,\theta _2=0.7$) -plots on the top-, and AMF-TASE$_4$ -RK4 (CMR), AMF-TASE$_4$ -RK4 (BFM), AMF-ROW-Kutta3/8 and AMF-TASE$_2$ -W4($\theta _1=0.5,\theta _2=0.7$) -plots on the bottom- when $N=40$ (plots on the left) and $N=80$ (plots on the right). Dashed straight lines with slopes two and three (plots on the top) and three and four (plots on the bottom), respectively, are included to compare the temporal orders of convergence

6.3 A 3D Linear Parabolic PDE with Variable Coefficients

We finally consider the 3D pure diffusion PDE (47) with $d_{ij}=a_i=0$, for $1\le i<j\le 3$, and variable diffusion coefficients $d_{ii}=d_{ii}(x_1,x_2,x_3)$, $1\le i\le 3$, given by $d_{11}=(1+x_1x_2x_3)^2,$ $d_{22}=e^{x_1-2x_2+3x_3}$ and $d_{33}=(1+x_1^2)e^{-x_2^2 x_3}$ (see [10]). Again $g(t,\textbf{x})$ is chosen in such way that (48), now with $\beta =2/7$, is the exact solution of (47). The MOL approach with second order central differences is applied on a uniform grid with mesh-width $h=\Delta x_i=1/(N+1)$, $1\le i\le 3$, for $N=40$ and $N=80$, whereas the AMF-TASE W-methods are applied with fixed stepsize $\tau =2^{-j}$, $2\le j\le 16$. As in Subsection 6.1, the AMF approach is exact. In this case, the obtained errors for each method when $N=40$ and $N=80$ are very similar and, therefore, we only present below in Fig. 7 the results corresponding to $N=80$. It can be observed that the methods AMF-TASE$_3$-RK3 (CMR), AMF-TASE$_3$ -RK3 (BFM), AMF-ROW3b and AMF-TASE$_2$ -W3($\theta _1=0.5,\theta _2=0.7$) perform with order three (see plot on the left). For the methods AMF-TASE$_4$ -RK4 (CMR), AMF-TASE$_4$ -RK4 (BFM), AMF-ROW-Kutta3/8 and AMF-TASE$_2$ -W4($\theta _1=0.5,\theta _2=0.7$), the observed order (see plot on the right) is between three and four when reaching the double precision limit in the calculations.

Fig. 7

Stepsize vs Error in the $\ell _\infty$-norm for the time integration of the semi-discrete 3D variable coefficient linear PDE of Subsection 6.3 with the methods AMF-TASE$_3$ -RK3 (CMR), AMF-TASE$_3$ -RK3 (BFM), AMF-ROW3b and AMF-TASE$_2$ -W3($\theta _1=0.5,\theta _2=0.7$) -plot on the left-, and AMF-TASE$_4$ -RK4 (CMR), AMF-TASE$_4$ -RK4 (BFM), AMF-ROW-Kutta3/8 and AMF-TASE$_2$ -W4($\theta _1=0.5,\theta _2=0.7$) -plot on the right- when $N=80$. Similar plots are obtained when $N=40$. Dashed straight lines with slope three (plot on the left), and three and four (plot on the right), respectively, are included to compare the temporal orders of convergence

7 Proofs of Theorems 3, 5 and 6

7.1 Proof of Theorem 3

The proof follows along the lines of [16, Theorem 2.11, pp. 151–152] for RK methods, [17, Theorems 7.3 and 7.4, pp. 106–107] for Rosenbrock methods and [17, Theorem 7.7, p. 116] for W-methods.

Associated to the set $\mathbb {T\widehat{W}}^{(d)}_q$, let us first introduce the set $L\mathbb {T\widehat{W}}^{(d)}_q$ of labelled rooted trees whose vertices are labelled monotonically taking into account the order q of the tree and the weight of each vertex (see, e.g., [16, p. 146] in the case of RK methods), in such a way that each tree $\mathcal {T}\in \mathbb {T\widehat{W}}^{(d)}_q$ corresponds exactly to $\alpha (\mathcal {T})$ labelled trees in $L\mathbb {T\widehat{W}}^{(d)}_q$. Then, to prove the thesis, it is sufficient to show that for the method (25)

$$\begin{aligned} \tilde{K}_{i{|\tau =0}}^{(q)}= \textstyle \sum _{\mathcal {T}\in L\mathbb {T\widehat{W}}^{(d)}_q} \gamma (\mathcal {T}) \Phi _i(\mathcal {T})F(\mathcal {T})(y_n), \quad i=1,\ldots ,s. \end{aligned}$$

(50)

Indeed, (36) would follow from (50) comparing the series expansions of the exact and numerical solutions as, e.g., in [17, Theorems 7.3 and 7.4, pp. 106–107].

We proceed by induction on q. For $q=1$, we have $\tilde{K}_{i{|\tau =0}}^{(1)}=f(y_n)=\gamma (\mathcal {T}_0) \Phi _i(\mathcal {T}_0) F(\mathcal {T}_0)(y_n)$. Let us then assume that (50) is true up to the derivative of order $q-1$ $(q\ge 2)$ and compute $\tilde{K}_{i{|\tau =0}}^{(q)}$ from (27).

For the first addend of $\tilde{K}_{i{|\tau =0}}^{(q)}$ in (27), we compute the $(q-1)$-th derivative using the Faà di Bruno’s formula [16, Lemma 2.8, p. 150], which reads

$$\begin{aligned}{}[f(u_i)]^{(q-1)}_{|\tau =0}=\textstyle \sum _{\mathcal {U} \in LS_q} \sum _{\begin{array}{c} \delta _1,\ldots ,\delta _l\ge 1\\ \delta _1+\cdots +\delta _l=q-1 \end{array}} f^{(l)}(u_i)(u^{(\delta _1)}_{i},\ldots ,u^{(\delta _l)}_{i})_{|\tau =0}, \end{aligned}$$

(51)

where $LS_q$ is the set of special labelled trees of order q having no ramifications except at the root (which is a black vertex). For $\mathcal {U} \in LS_q$ in (51), l is the number of branches leaving the root and $\delta _1,\ldots ,\delta _l\ge 1$ are the number of nodes in each of these branches, such that $q=1+\delta _1+\cdots +\delta _l$ (see, e.g., [16, Definition 2.7, p. 150]). The expressions of $u^{(\delta _m)}_{i{|\tau =0}}, \ m=1,\ldots ,l$, can be computed from (27). For the second addend of $\tilde{K}_{i{|\tau =0}}^{(q)}$ in (27), the $(q-1)$-th derivative is directly computed from Leibniz product rule.

Through the inductive hypothesis, we get using (50) that

$$\begin{aligned} \begin{aligned} \tilde{K}_{i{|\tau =0}}^{(q)}=&q \sum _{\mathcal {U}\in LS_q} \sum _{\begin{array}{c} \delta _1,\ldots ,\delta _l\ge 1\\ \delta _1+\cdots +\delta _l=q-1 \end{array}} \sum _{\mathcal {T}_1\in L\mathbb {T\widehat{W}}^{(d)}_{\delta _1}} \cdots \sum _{\mathcal {T}_l \in L\mathbb {T\widehat{W}}^{(d)}_{\delta _{l}}} \gamma (\mathcal {T}_1) \cdots \gamma (\mathcal {T}_l) \\&\cdot \sum _{j_1} {\tilde{a}_{i,j_1}} \Phi _{j_1}(\mathcal {T}_1) \cdots \sum _{j_l} {\tilde{a}_{i,j_l}} \Phi _{j_l}(\mathcal {T}_l) \, f^{(l)} (y_n) (F(\mathcal {T}_1), \ldots ,F(\mathcal {T}_l))\\&+ q\sum _{\nu =1}^{\min \{q-1,d\}} \sum _{\mathcal {T}_1\in L\mathbb {T\widehat{W}}^{(d)}_{q-\nu }} \displaystyle \frac{(q-1)!}{ (q-\nu )!} \gamma (\mathcal {T}_1) \sum _{j} (-\theta _i)^{\nu -1} \gamma _{i,j} \Phi _j(\mathcal {T}_1)\, \widehat{W}_\nu F(\mathcal {T}_{1})(y_n). \end{aligned} \end{aligned}$$

(52)

Similarly as in the proof of [17, Theorem 7.3, p. 106], the following one-to-one correspondences hold true:

The expression (50) then follows from (52) using (32)–(33)–(34)–(35) and that the sets of trees on the left of (53) cover all $L\mathbb {T\widehat{W}}^{(d)}_q$. $\square$

7.2 Proof of Theorem 5

The proof proceeds by induction on $p\ge 2$. For $p=2$, with the tree $\widehat{\mathcal {T}}=\mathcal {T}_0$, the proof is immediate since, from (37)–(38), the corresponding order condition is just $\eta _1 {\tilde{b}}^\top \tilde{\Gamma }_{\mathbb {1}} \textbf{1}_s=0.$ Let us assume that the statement holds up to order $p-1$. We consider an integer $k\ge 1$ and a tree $\widehat{\mathcal {T}}$ ($\widehat{\mathcal {T}}=\mathcal {T}_0$ or $\widehat{\mathcal {T}}=[\mathcal {T}_1,\ldots ,\mathcal {T}_q]_\bullet \in \mathbb{T}\mathbb{W}$, $q\ge 1$) fulfilling $k+\rho (\widehat{\mathcal {T}})\le p$. For partitions of k given by the integers $1\le n_1,\ldots ,n_j\le k$ with $n_1+\cdots +n_j=k$ ($j=1,\ldots ,k$), we distinguish two situations:

• a) there exist indexes $i, l \in \{1,\ldots ,j\}$ such that $n_i\ne n_l$, or

• b) $n_i=\tilde{n}$, for all $i=1,\ldots ,j$.

First, for a partition in the situation of a), if $\eta _{n_1}\cdots \eta _{n_j}=0$, then (39) automatically follows. Therefore, we can assume that $\eta _{n_1}\cdots \eta _{n_j}\ne 0$. For such partition, we can also assume without loss of generality that $1\le n_1\le \ldots \le n_j\le k$. Consequently, we have that $n_1+\overset{(j)}{\ldots }+n_1=jn_1<k.$ For the integer $\tilde{k}:=j\cdot n_1$, let us consider the particular partition given by $(n_1, \overset{(j)}{\ldots }, n_1)$ and the tree , for which $\rho ( \tilde{\mathcal { T}})=\tilde{k}+\rho (\widehat{\mathcal {T}})<k+\rho (\widehat{\mathcal {T}})\le p$. Therefore, through the inductive hypothesis we get $\eta _{n_1}^j \tilde{b}^\top \tilde{\Gamma }^{j-1}\tilde{\Gamma }_{\mathbb {1}} \phi (\widehat{\mathcal {T}})=0.$ From $\eta _{n_1}\ne 0$, $\tilde{b}^\top \tilde{\Gamma }^{j-1} \tilde{\Gamma }_{\mathbb {1}} \phi (\widehat{\mathcal {T}})=0$, proving the identity of the right of (39) in case a).

We now proceed with the partitions of k under the situation of b). Observe that the order condition on the left of (39) now reduces to (with $j=k/\tilde{n}$)

$$\begin{aligned} \textstyle \sum _{\tilde{n}|k} \eta _{\tilde{n}}^{k/\tilde{n}}\; \tilde{b}^\top \tilde{\Gamma }^{k/\tilde{n}-1} \tilde{\Gamma }_{\mathbb {1}} \phi (\widehat{\mathcal {T}})=0, \end{aligned}$$

(54)

where $\sum _{\tilde{n}|k}$ indicates the sum over all the $\tilde{n}=1,\ldots ,k$ such that $k/\tilde{n}$ is an integer. Again, the proof of (39) would follow immediately if $\eta _{\tilde{n}}=0$ for all $\tilde{n}$ divisor or k. Otherwise, by defining

$$\begin{aligned} \tilde{n}^*:=\min \Bigl \{\tilde{n}:\; k/\tilde{n} \in \mathbb {N},\; \eta _{\tilde{n}}\ne 0\Bigr \}, \end{aligned}$$

(55)

we can write the identity in (54) as

$$\begin{aligned} \eta _{\tilde{n}^*}^{k/\tilde{n}^*} \;\tilde{b}^\top \tilde{\Gamma }^{k/\tilde{n}^*-1} \tilde{\Gamma }_{\mathbb {1}} \phi (\widehat{\mathcal {T}})+ \textstyle \sum _{\begin{array}{c} \tilde{n}|k\\ \tilde{n}>\tilde{n}^* \end{array}} \eta _{\tilde{n}}^{k/\tilde{n}} \;\tilde{b}^\top \tilde{\Gamma }^{k/\tilde{n}-1} \tilde{\Gamma }_{\mathbb {1}} \phi (\widehat{\mathcal {T}})=0. \end{aligned}$$

(56)

For each $\tilde{n}>\tilde{n}^*$, with $k/\tilde{n} \in \mathbb {N}$, consider the particular partition $(\tilde{n}^*,\overset{(k/\tilde{n})}{\ldots },\tilde{n}^*)$ of $\tilde{n}^*\cdot \left( k/\tilde{n}\right)$, related to the tree , for which $\rho (\tilde{\mathcal {T}})= \tilde{n}^*\cdot \left( k/\tilde{n}\right) +\rho (\widehat{\mathcal {T}})<k+\rho (\widehat{\mathcal {T}})\le p$. Thus, through the inductive hypothesis we know that

$$\begin{aligned} \eta _{\tilde{n}^*}^{k/\tilde{n}} \;\tilde{b}^\top \tilde{\Gamma }^{k/\tilde{n}-1} \tilde{\Gamma }_{\mathbb {1}} \phi (\widehat{\mathcal {T}})=0, \qquad \forall \tilde{n}>\tilde{n}^*,\;\; k/\tilde{n}\in \mathbb {N}, \end{aligned}$$

where, from (55), $\eta _{\tilde{n}^*}\ne 0$. Therefore, $\tilde{b}^\top \tilde{\Gamma }^{k/\tilde{n}-1} \tilde{\Gamma }_{\mathbb {1}} \phi (\widehat{\mathcal {T}})=0$, for all $\tilde{n}>\tilde{n}^*$, with $k/\tilde{n}\in \mathbb {N}$. This inserted in (56) also implies that $\tilde{b}^\top \tilde{\Gamma }^{k/\tilde{n}^*-1} \tilde{\Gamma }_{\mathbb {1}} \phi (\widehat{\mathcal {T}})=0$. Hence, each addend of (54) is zero and the statement (39) follows also for partitions of k under the situation of b). $\square$

Remark 6

As underlined in Remark 4, from Theorem 5 we get that in the order conditions (37) of TASE$_r$ W-methods for trees $\mathcal {T}\in \mathbb{T}\mathbb{W}{\setminus } \mathbb {T}$, $\rho (\mathcal {T})\le p$, each addend taken individually is zero in the situation of the last three statements of (38). This observation is immediate for the last two statements. The second statement requires further explanation. In the second statement of (38), more general trees of the form $\mathcal {T}=[\mathcal {T}_1,\ldots ,\mathcal {T}_q]_\bullet ,$ $\mathcal {T}_1,\ldots ,\mathcal {T}_q\in \mathbb{T}\mathbb{W}$ ($q\ge 1$), are considered, for which the corresponding order condition takes the expression $\tilde{b}^\top \phi (\mathcal {T})= \tilde{b}^\top \left[ (\tilde{A}\phi (\mathcal {T}_1)) \varvec{*} \cdots \varvec{*} (\tilde{A}\phi (\mathcal {T}_q))\right] .$ In the set $\mathbb{T}\mathbb{W}\setminus \mathbb {T}$, the only tree of this kind with lowest order ($p=3$) is given by , for which the corresponding order condition $\eta _1\cdot \tilde{b}^\top \tilde{A}\rho =0$ (see Table 3) just contains one addend. Proceeding by induction on the order of the tree, when $\mathcal {T}$ contains a subtree of the form , with $k\ge 1$ and $\widehat{\mathcal {T}} \in \mathbb{T}\mathbb{W}$ with a black root, we can proceed exactly with the same order reduction strategy, based on merging consecutive white nodes, as in the proof of Theorem 5. This order reduction approach is also valid for general trees containing a chain of k consecutive white nodes. Observe that when, for a given tree, the length of the largest chain of white consecutive nodes is $k=1$, then the corresponding order condition just contains one addend (with a factor $\eta _1$ appearing as many times as the number of white nodes in the tree).

7.3 Proof of Theorem 6

The proof proceeds similarly as in the proof of Theorem 4 for TASE W-methods given in [11, Theorem 2]. First, it is not difficult to check that (41) and (42) follow from (40) taking into account (9) and that $\widehat{\tilde{A}}=\tilde{A} \otimes \Omega$ and $\widehat{\tilde{b}}^\top =\tilde{b}^\top \otimes \omega ^\top =(\tilde{b}^\top \otimes \omega ^\top )(I_s\otimes \Omega )$. Let us then prove (40) with the corresponding functions $\Phi (\mathcal {T})$ in (35) and $\varphi (\mathcal {T})$ in (43).

i) For trees $\mathcal {T}\in \mathbb {T}$ (with only black vertices), the proof of the first and second statement in (43) follows exactly as in the item i) of the proof of [11, Theorem 2].

ii) In order to prove (40) for the third statement in (43), let us consider a tree , with $\nu _1+\cdots +\nu _j=k$, $1\le j\le k.$ Considering the definition of $\Phi (\mathcal {T})$ in (35) and the method coefficients (10) and (8), it follows that $\Phi (\mathcal {T})=G_{k,j} (\textbf{1}_{s} \otimes \textbf{1}_{r})$, where the matrix $G_{k,j}$ is given by

$$\begin{aligned} G_{k,j}:=(-1)^{k-j} \textstyle \prod _{i=1}^{j} \Bigl ((I_s \otimes \Theta ^{\nu _i-1} ) \widehat{\Gamma }\Bigr )=(-1)^{k-j} \prod _{i=1}^{j} \Bigl (I_s \otimes \Theta ^{\nu _i} + \tilde{\Gamma }\otimes \Theta ^{\nu _i} \Omega \Bigr ). \end{aligned}$$

(57)

Expanding the latter product we get

$$\begin{aligned} \begin{aligned} G_{k,j}=&(-1)^{k-j} \textstyle \sum _{i=0}^{j}\;\; \sum _{\begin{array}{c} m_1,\ldots ,m_j \in \{0,1\} \\ m_1+\cdots +m_j=i \end{array} } \tilde{\Gamma }^i \otimes (\Theta ^{\nu _1}\Omega ^{m_1}) \cdots (\Theta ^{\nu _j}\Omega ^{m_j}) \\ =&(-1)^{k-j} \textstyle \sum _{i=0}^{j-1}\;\; \sum _{\begin{array}{c} m_1,\ldots ,m_{j-1} \in \{0,1\} \\ m_1+\cdots +m_{j-1}=i \end{array} } \tilde{\Gamma }^i \otimes (\Theta ^{\nu _1}\Omega ^{m_1}) \cdots (\Theta ^{\nu _{j-1}}\Omega ^{m_{j-1}}) \Theta ^{\nu _j} \\ +&(-1)^{k-j} \textstyle \sum _{i=1}^{j}\;\; \sum _{\begin{array}{c} m_1,\ldots ,m_{j-1} \in \{0,1\} \\ m_1+\cdots +m_{j-1}=i-1 \end{array} } \tilde{\Gamma }^i \otimes (\Theta ^{\nu _1}\Omega ^{m_1}) \cdots (\Theta ^{\nu _{j-1}}\Omega ^{m_{j-1}}) \Theta ^{\nu _j} \Omega . \end{aligned} \end{aligned}$$

(58)

Since $\Omega \Theta ^n \textbf{1}_r=\eta _n \textbf{1}_r$, for all $n\ge 1$, from (58) it holds for all vectors $v\in \mathbb {R}^s$ that

$$\begin{aligned} \begin{aligned} (I_s\otimes \Omega )G_{k,j}(v\otimes \textbf{1}_r)=&\biggl ((-1)^{k-j}\textstyle \sum _{i=0}^{j-1} \Bigl (\textstyle \sum _{\begin{array}{c} 1\le l_1,\ldots ,l_{i+1}\le j \\ l_1+\cdots +l_{i+1}=j \end{array} } \eta _{p_1}\cdots \eta _{p_{i+1}}\Bigr ) \tilde{\Gamma }^i v \biggr ) \otimes \textbf{1}_r\\+&\biggl ((-1)^{k-j}\textstyle \sum _{i=1}^{j} \Bigl (\sum _{\begin{array}{c} 1\le l_1,\ldots ,l_{i}\le j \\ l_1+\cdots +l_{i}=j \end{array} } \eta _{p_1}\cdots \eta _{p_{i}}\Bigr ) \tilde{\Gamma }^i v \biggr ) \otimes \textbf{1}_r, \end{aligned} \end{aligned}$$

where $p_m$, $m=1\ldots ,i$, fulfil (44). At the end we have, for all $v \in \mathbb {R}^s$,

$$\begin{aligned} \begin{aligned} (I_s\otimes \Omega )G_{k,j}(v\otimes \textbf{1}_r)&= \biggl ((-1)^{k-j}\textstyle \sum _{i=1}^{j} \Bigl (\sum _{\begin{array}{c} 1\le l_1,\ldots ,l_{i}\le j \\ l_1+\cdots +l_{i}=j \end{array} } \eta _{p_1}\cdots \eta _{p_{i}}\Bigr ) \tilde{\Gamma }^{i-1}(\tilde{\Gamma }+ I_s) v \biggr ) \otimes \textbf{1}_r. \end{aligned} \end{aligned}$$

(59)

This proves (40) for by taking $v=\textbf{1}_s$.

iii) It remains to prove (40) in the situations of the second and fourth statement in (43), which is done next recursively. To this aim, let $\mathcal {T}_1,\ldots ,\mathcal {T}_q\in \mathbb {T\widehat{W}}$ satisfy $(I_s\otimes \Omega ) \Phi (\mathcal {T}_i)=\varphi (\mathcal {T}_i)\otimes \textbf{1}_r$ and $\widehat{\tilde{A}}\Phi (\mathcal {T}_i)=\tilde{A}\varphi (\mathcal {T}_i)\otimes \textbf{1}_r$, $i=1,\ldots ,q$. We then have for $\mathcal {T}=[\mathcal {T}_1,\ldots ,\mathcal {T}_q]_\bullet$ that

$$\begin{aligned} \begin{aligned} (I_s\otimes \Omega )\Phi (\mathcal {T})&=(I_s\otimes \Omega )\Bigl (((\tilde{A}\varphi (\mathcal {T}_1)) \varvec{*} \cdots \varvec{*} (\tilde{A}\varphi (\mathcal {T}_q)))\otimes \textbf{1}_r\Bigr )=\varphi (\mathcal {T})\otimes \textbf{1}_{r}, \end{aligned} \end{aligned}$$

and for , with $\nu _1+\cdots +\nu _j=k$, $1\le j\le k$, and $\widehat{\mathcal {T}}=[\mathcal {T}_1,\ldots ,\mathcal {T}_q]_\bullet$,

$$\begin{aligned} \begin{aligned} (I_s\otimes \Omega )\Phi (\mathcal {T})&=(I_s\otimes \Omega )G_{k,j}\Phi (\widehat{\mathcal {T}})\\ {}&=(I_s\otimes \Omega )G_{k,j}\Bigl ((\widehat{\tilde{A}}\Phi (\mathcal {T}_1)) \varvec{*} \cdots \varvec{*} (\widehat{\tilde{A}}\Phi (\mathcal {T}_q))\Bigr ) =(I_s\otimes \Omega )G_{k,j}\bigl (\varphi (\widehat{\mathcal {T}})\otimes \textbf{1}_{r}\bigr ), \end{aligned} \end{aligned}$$

using (57) in the first identity. The proof finishes inserting $v=\varphi (\widehat{\mathcal {T}})$ in (59). $\square$

8 Concluding Remarks

The application of AMF [20] in combination with linearly implicit ODE methods [17, 26] provides an efficient way of numerically solving large semi-discrete parabolic PDE problems in several spatial dimensions. Interesting particular subclasses of such linearly implicit methods are the so-called W-methods [25] and the TASE W-methods recently introduced in [11] aimed at reducing the computational cost of the TASE-RK methods in [2] and [5]. While for the class of AMF W-methods [9] it trivially holds that the AMF retains the temporal consistency order of the underlying W-method, this property needs a more thorough analysis for the newly introduced AMF-TASE W-methods. Indeed, for these latter methods we have described what additional order conditions must be fulfilled. The parallel structure of the AMF-TASE W-methods allows to show that these additional order conditions are automatically fulfilled from the order conditions of the underlying TASE W-method, and thus the AMF approach also retains the order of consistency. This is numerically confirmed on a nonlinear diffusion–reaction problem, a linear advection–diffusion PDE with mixed derivatives and a pure diffusion linear PDE with variable coefficients in three spatial dimensions. We underline that the good numerical performances were achieved through a sequential implementation of the AMF-TASE W-methods, thus not exploiting the intrinsic parallelism of the new approach, which could be the subject of future research. Aspects such as linear stability properties of the AMF-TASE W-methods and its convergence to the PDE solution in terms of the mesh-width require additional analysis. In this regard, as for AMF W-methods (see, e.g, [7, 8, 10]), the PDE order of convergence (i.e., the order of convergence when both the stepsize $\tau$ and the mesh-width h tend to zero with $\tau =\mathcal {O}(h)$) of AMF-TASE W-methods is expected to be $\min \{p,3.25\}$ (being p the temporal order of consistency of the method) for the case of linear parabolic problems, the Euclidean norm and time independent Dirichlet boundary conditions (for time dependent boundary conditions the PDE order of convergence is expected to be $\min \{p,2\}$).