# Runge–Kutta time semidiscretizations of semilinear PDEs with non-smooth data

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## Abstract

We study semilinear evolution equations \( \frac{\mathrm dU}{\mathrm dt}=AU+B(U)\) posed on a Hilbert space \(\mathcal Y\), where *A* is normal and generates a strongly continuous semigroup, *B* is a smooth nonlinearity from \(\mathcal Y_\ell = D(A^\ell )\) to itself, and \(\ell \in I \subseteq [0,L]\), \(L \ge 0\), \(0,L \in I\). In particular the one-dimensional semilinear wave equation and nonlinear Schrödinger equation with periodic, Neumann and Dirichlet boundary conditions fit into this framework. We discretize the evolution equation with an A-stable Runge–Kutta method in time, retaining continuous space, and prove convergence of order \(O(h^{p\ell /(p+1)})\) for non-smooth initial data \(U^0\in \mathcal Y_\ell \), where \(\ell \le p+1\), for a method of classical order *p*, extending a result by Brenner and Thomée for linear systems. Our approach is to project the semiflow and numerical method to spectral Galerkin approximations, and to balance the projection error with the error of the time discretization of the projected system. Numerical experiments suggest that our estimates are sharp.

## Mathematics Subject Classification

65J08 65J15 65M12 65M15## 1 Introduction

*A*is a normal linear operator that generates a strongly continuous semigroup, and that

*B*is smooth on a scale of Hilbert spaces \(\{\mathcal Y_\ell \}_{\ell \in I}\), \(I \subseteq [0,L]\), \(0,L \in I\), as detailed in condition (B) below. Here \(\mathcal Y_\ell =D(A^\ell )\subseteq \mathcal Y\), \(\ell \ge 0\). Note that condition (B) depends on both, the smoothness properties of the nonlinearity

*B*(

*U*) and the boundary conditions. Under these assumptions the class of equations we consider includes the semilinear wave equation and the nonlinear Schrödinger equation in one spatial dimension with periodic, Neumann and Dirichlet boundary conditions (see Examples 2.3–2.8 below). For an example in three space dimensions see Example 5.4. We discretize (1.1) in time by an

*A*-stable Runge Kutta method; the condition of

*A*-stability ensures that the numerical method is well-defined on \(\mathcal Y\), and is satisfied by a large class of methods including the Gauss–Legendre collocation methods.

Discretizing in time while retaining a continuous spatial parameter means that we consider the numerical method as a nonlinear operator on the infinite dimensional space \(\mathcal Y\). This leads to several technicalities, in particular existence results for the numerical method \(\Psi ^h\) as well as the semiflow \(\Phi ^t\) and regularity of solutions in both cases are required to ensure convergence results analogous to the finite dimensional case. In [15], existence and regularity of the semiflow of (1.1) on a scale of Hilbert spaces, corresponding results for the numerical method, and full order convergence of the time semidiscretization for sufficiently smooth data are studied in detail. We review the relevant results in Sects. 2 and 3.

In this paper we consider the effect of non-smooth data on the order of convergence of the time semidiscretization in this setting. We consider an *A*-stable Runge–Kutta method of classical order *p* applied to the problem (1.1) with initial data \(U^0\in \mathcal Y_\ell \), \(\ell \in I\). The main result we give here, Theorem 5.3, shows that we can expect order of convergence \(\mathcal O(h^q)\) where \(q(\ell ) = p\ell /(p+1)\) for \(0\le \ell < p+1\). This corresponds closely with numerical observation, cf. Fig. 1. Given a time \(T>0\) we prove the above order of convergence for the time-semidiscretization up to time *T* for any solution *U*(*t*) of (1.1) with a given \(\mathcal Y_\ell \) bound. Here \(\ell >0\) is such that \(\ell -k \in I\) for \(k=1,\ldots , \lfloor \ell \rfloor \) (the greatest integer \(\le \ell \)). It is shown in [15] that for \(\ell \ge p+1\) we have full order of convergence \(\mathcal O(h^p)\).

The reduction in order of the method from *p* to *q* for \(\ell <p+1\) is caused by the occurrence of unbounded operators in the Taylor expansion of the one-step error coefficient. Our approach is to apply a spectral Galerkin approximation to the semiflow of the evolution equation (1.1), and to discretize the projected evolution equation in time. This allows us to bound the size of the local error coefficients in terms of the accuracy of the projection. By balancing the projection error with the growth of the local error coefficients we obtain the estimates of our main result, Theorem 5.3.

Related results include those of Brenner and Thomée [3], who consider linear evolution equations \(\dot{U}=AU\) in a more general setting, namely posed on a Banach space \(\mathcal X\), where *A* generates a strongly continuous semigroup \(e^{tA}\) on \(\mathcal X\). They show \(O(h^q)\) convergence of A-acceptable rational approximations of the semigroup for non-smooth initial data \(U^0\in D(A^\ell )\), \(\ell = 0,\ldots ,p+1\), with \(q=q(\ell )= p \ell /(p+1)\) as above, if \(\ell > (p+1)/2\) (when \(\ell \le (p+1)/2\) they prove convergence with order \(q(\ell )< p \ell /(p+1)\)). Kovács [9] generalizes this result to certain intermediate spaces with arbitrary \(\ell \in [0,p+1]\) and also provides sufficient conditions for when \(q=q(\ell )= p \ell /(p+1)\) for all \(\ell \in [0,p+1]\) (which are satisfied in our setting).

For splitting methods, where the linear part of the evolution equation is evaluated exactly, a higher order of convergence has been obtained for specific choices of \(\ell \) and specific evolution equations in [6] and [13], see also Example 5.4 below. While splitting methods are very effective for simulating evolution equations for which the linear evolution \(\mathrm e^{tA}\) can easily be computed explicitly, Runge–Kutta methods are still a good choice when an eigen-decomposition of *A* is not available, for example for the semilinear wave equation in an inhomogeneous medium, see Example 2.7. Moreover, the simplest example of a Gauss–Legendre Runge–Kutta method, the implicit mid point rule, appears to have some advantage over split step time-semidiscretizations for the computation of wave trains for nonlinear Schrödinger equations because the latter introduce an artificial instability [18].

For Runge–Kutta time semidiscretizations of dissipative evolution equations, where *A* is sectorial, a better order of convergence can be obtained, see [10] for the linear case and [11, 12] and references therein for the semilinear case.

Note that our approach is different from the approach of [11, 12]. In [11, 12] some smoothness of the continuous solution is assumed and from that a (fractional) order of convergence is obtained, using the variation of constants formula. The order of convergence obtained in [11, 12] is in general lower than in the linear case (where full order of convergence is obtained in the parabolic case [10]), but no extra assumptions on the nonlinearity *B*(*U*) of the PDE are made. In particular in [12, Theorems 4.1 and 4.2] the existence of \((p_s+2)\) time derivatives of the continuous solution *U*(*t*) of a semilinear parabolic PDE (1.1) is assumed, where \(p_s\) is the stage order of the method. This assumption is then used to estimate the error of the numerical approximation of the inhomogenous part of the variation of constants formula. Here the stage order \(p_s\) comes into play. Note that if the nonlinearity *B*(*U*) of the evolution equation (1.1) only satisfies the standard assumption rather than our assumption (B), i.e., is smooth on \(\mathcal Y\) only (so that the Hilbert space scale is trivial with \(L=0\)) then the existence of \(U'(t)\) can be guaranteed for \(U^0 \in \mathcal Y_1\) by semigroup theory [17], but it is not clear whether higher order time derivatives of the solution *U*(*t*) of (1.1) exist as assumed in [12]—therefore in [12] also time-dependent perturbations of (1.1) are considered. In this paper we instead take the approach of making assumptions (namely condition (B) on the nonlinearity *B*(*U*) of the evolution equation and the condition that \(U^0 \in \mathcal Y_\ell \)) which are straightforward to check and guarantee the existence of the time derivatives of the continuous solution *U*(*t*) up to order \(k\le \ell \). We then obtain an order of convergence \(O(h^{p\ell /(p+1)})\) of the Runge–Kutta discretization which is identical to the order of convergence in the linear case [3, 9]. In [11, Theorem 2.1] some smoothness of the inhomogeneity of the PDE is obtained from the smoothing properties of parabolic PDEs, and this is used to prove an order of convergence \(h\log h\), without the assumption of the existence of higher time derivatives of the continuous solution *U*(*t*). Here we do not consider parabolic PDEs, so that we cannot use this strategy.

Alonso-Mallo and Palencia [2] study Runge–Kutta time discretizations of inhomogeneous linear evolution equations where the linear part creates a strongly continuous semigroup. Similarly as in [12] they obtain an order of convergence depending on the stage order \(p_s\) of the Runge–Kutta method. They assume the continuous solution *U*(*t*) to be \((p+1)\)-times differentiable in *t*, but in their context the condition \(U(t) \in D(A^{p-p_s})\), where *p* is the order of the numerical method, is in general not satisfied due to the inhomogeneous terms in the evolution equation, and this leads to a loss in the order of convergence compared to our results. Note that in our setting, due to our condition (B) on the nonlinearity, provided \(U(0) \in \mathcal Y_{p+1}\) we have \(U(t) \in D(A^{p+1}) = \mathcal Y_{p+1}\) and *U*(*t*) is \(p+1\) times differentiable in *t* (in the \(\mathcal Y\) norm) and so we get full order of convergence in this case (see [15]). Calvo et al. [4] study Runge–Kutta quadrature methods for linear evolution equations \(\dot{U}(t) = A(t)U(t)\) which are well-posed and prove full order convergence if the continuous solution *U*(*t*) has \(p+1\) time derivatives; they also obtain fractional orders of convergence as in [3] for solutions \(U(t) \in \mathcal Y_\ell \) with \(\ell <p+1\).

We proceed as follows: in Sect. 2 we introduce the class of semilinear evolution equations that we consider in this paper, give some examples, review existence and regularity results of [15, 17] for the semiflow, and adapt them to the case of non-integer \(\ell \). In Sect. 3 we introduce a class of *A*-stable Runge–Kutta methods. We review existence and regularity of these methods when applied to the semilinear evolution equation (1.1) and a convergence result for sufficiently smooth initial data from [15]. In Sect. 4 we study the stability of the semiflow and numerical method under spectral Galerkin truncation, and establish estimates for the projection error. Lemma 4.2 and 4.3 are established in [16] for integer values of \(\ell \); for completeness we review the proofs, which also work for non-integer \(\ell \). In Sect. 5 we prove our main result on convergence of *A*-stable Runge–Kutta discretizations of semilinear evolution equations for non-smooth initial data.

## 2 Semilinear PDEs on a scale of Hilbert spaces

In this section we introduce a suitable functional setting for the class of equations we subsequently study. We review results from [15, 17] on the local well-posedness and regularity of solutions of (1.1) and give examples.

*R*around \(U^0\) in \(\mathcal X\). We make the following assumptions on the semilinear evolution equation (1.1):

- (A)
*A*is a normal linear operator on \(\mathcal Y\) that generates a strongly continuous semigroup of linear operators \(e^{tA}\) on \(\mathcal Y\) in the sense of [17].

*A*to \({\text {spec}}\ (A) \cap \mathcal B^m_\mathbb C(0)\), let \(\mathbb Q_m={\text {id}}-\mathbb P_m\) and set \(\mathbb P=\mathbb P_1\), \(\mathbb Q= {\text {id}}- \mathbb P\). We endow \(\mathcal Y_\ell \) with the inner product

### *Remark 2.1*

*B*, we introduce the following notation: for Banach spaces \(\mathcal X\), \(\mathcal Z\), we denote by \(\mathcal E^i(\mathcal X,\mathcal Z)\) the space of

*i*-multilinear bounded mappings from \(\mathcal X\) to \(\mathcal Z\). For \(\mathcal U\subseteq \mathcal X\) we write \(\mathcal C_{{\text {b}}}^k(\mathcal U,\mathcal Z)\) to denote the set of

*k*times continuously differentiable functions \(F :{\text {int}}\mathcal U\rightarrow \mathcal Z\) such that

*F*and its derivatives \(\mathrm D^i F\) are bounded as maps from the interior \({\text {int}}\mathcal U\) of \(\mathcal U\) to \(\mathcal E^i(\mathcal X,\mathcal Z)\) and extend continuously to the boundary of \({\text {int}}\mathcal U\) for \(i \le k\). We set \(\mathcal C_{{\text {b}}}(\mathcal U,\mathcal Z)= \mathcal C_{{\text {b}}}^0(\mathcal U,\mathcal Z)\). Note that if \(\dim \mathcal X=\infty \), there are examples of continuous functions \(F:\mathcal U\rightarrow \mathcal Z\) where \(\mathcal U\) is closed and bounded, which do not lie in \(\mathcal C_{{\text {b}}}(\mathcal U,\mathcal Z)\), see e.g. [15, Remark 2.3]. In the following for \(\ell \in \mathbb R\) let \(\lfloor \ell \rfloor \) be the largest integer less than or equal to \(\ell \) and \(\lceil \ell \rceil \) be the smallest integer greater or equal to \(\ell \). Moreover for \(R>0\) and \(\ell \ge 0\) we abbreviate

*B*(

*U*) of (1.1).

- (B)
There exists \(L\ge 0\), \(I \subseteq [0,L]\), \(0,L \in I\), \(N\in \mathbb N\), \(N > \lceil L\rceil \), such that \(B\in \mathcal C_{{\text {b}}}^{N- \lceil \ell \rceil }(\mathcal B^R_\ell ;\mathcal Y_\ell )\) for all \(\ell \in I\) and \(R>0\).

### **Theorem 2.2**

*R*, \(\omega \) from (2.1), and the bounds afforded by assumption (B) on balls of radius

*R*.

### *Proof*

*N*derivatives in the first component. This proves statements (2.8a) and also \(\Phi (U) \in \mathcal C_{{\text {b}}}^k([0,T_*];\mathcal B_0^R)\) in the case \(k=0\).

For \(k \in \mathbb N\), \(k\le \ell \) it follows from the fact \(\ell \in I^-\) that the above argument applies with \(\mathcal Y\) replaced by \(\mathcal Y_{\ell -j}\), \(j=0,\ldots , k\). Hence there is some \(T_*>0\) such that \(\Phi \in \mathcal C_{{\text {b}}}(\mathcal B^{R/2}_{\ell -j}\times [0,T_*];\mathcal B^{R}_{\ell -j})\) for \(j=0,\ldots , k\). As detailed in [15] for \(U \in \mathcal B^{R/2}_{\ell }\) the *t* derivatives up to order *k* can then be obtained by implicit differentiation of \(\Pi (W(U,T),U,T)=W(U,T)\) with \(\Pi \) defined above which implies that \(\Phi (U) \in \mathcal C_{{\text {b}}}^k([0,T_*];\mathcal B^R_0)\) for \(k\le \ell \) with uniform bounds in \(U \in \mathcal B^{R/2}_{\ell }\). \(\square \)

Note that this theorem extends to mixed (*U*, *t*) derivatives which are, however, in general only strongly continuous in *t*, see [15] for details. For our purposes in this paper the above theorem is sufficient.

### *Example 2.3*

*Semilinear wave equation, periodic boundary conditions*) Consider the semilinear wave equation

*A*separates into \(2\times 2\) eigenvalue problems on each Fourier mode, and it is easy to see that the spectrum of

*A*is given by

*B*. We denote the Fourier coefficients of a function \(u \in \mathcal L^2([0,2\pi ];\mathbb R^d)\) by \(\hat{u}_k\), so that

*B*(

*U*) is analytic as map of \(\mathcal Y_{\ell }\) to itself for any \(\ell \ge 0\) and

*B*and its derivatives are bounded on balls around 0. Hence assumption (B) holds for any \(L\ge 0\) and \(N> \lceil L \rceil \) with \(I=[0,L]\).

### *Example 2.4*

(*Semilinear wave equation, non-analytic nonlinearity*) If \(V\in \mathcal C^{N+2}(\mathbb R)\) then (B) holds with \(I = [0,L]\) and \(\lceil L\rceil < N\). To see this note that Lemma 2.9 c) applied to \(f = V' \in \mathcal C^{N+1}(\mathbb R)\) ensures that \(f \in \mathcal C_{{\text {b}}}^{N-\lfloor \ell \rfloor }(\mathcal B_{\mathcal H_{\ell +1}}^R; \mathcal H_{\ell })\) for all \(R>0\) and therefore that (B) holds, noting that \(\mathcal Y_\ell \) is as in (2.15). Here we abbreviated \(\mathcal H_{\ell }:=\mathcal H_{\ell }([0,2\pi ];\mathbb R)\).

### *Example 2.5*

*Semilinear wave equation, Dirichlet boundary conditions*) When endowed with homogeneous Dirichlet boundary conditions \(u(t,0) = u(t,\pi )=0\) the linear part

*A*of the semilinear wave equation (2.11) still generates a unitary group. In this case we have \(\mathbb P_0=0\), \(A=\tilde{A}\), and

*V*is even may be relaxed to the requirement that \(V^{(2j+1)}(0)=0\) for \(0\le 2j\le L+\frac{1}{2}\).

### *Example 2.6*

*Semilinear wave equation, Neumann boundary conditions*) In the case of Neumann boundary conditions on \( [0,\pi ]\), the operator \(A=\tilde{A}\) from (2.12) is again skew-symmetric and has the same spectrum as in Example 2.3. In this case, \(\mathcal Y_\ell = \mathcal H_{\ell +1}^{{\text {nb}}}( [0,\pi ];\mathbb R) \times \mathcal H_{\ell }^{{\text {nb}}}( [0,\pi ];\mathbb R)\). Here \(\mathcal H_\ell ^{{\text {nb}}}([0,\pi ];\mathbb R) = D((-\Delta )^{\ell /2})\), where \(\Delta \) now denotes the Laplacian with Neumann boundary conditions. Due to [8]

*u*of order at most \(2j + 1\), so that the required boundary conditions for

*f*are satisfied. Hence Condition (B) is satisfied for any \(L\ge 0\) with \(I = [0,L] \).

### *Example 2.7*

*A semilinear wave equation in an inhomogeneous material*) Instead of (2.11), let us consider the non-constant coefficient semilinear wave equation

### *Example 2.8*

*Nonlinear Schrödinger equation*) Consider the nonlinear Schrödinger equation

*A*is given by

*A*is normal and generates a unitary group on \(\mathcal L^2([0,2\pi ];\mathbb C)\) and, more generally, on every \(\mathcal H_{\ell }([0,2\pi ];\mathbb C)\) with \(\ell \ge 0\).

By Lemma 2.9 a) below the nonlinearity *B*(*U*) defined in (2.17) is analytic as map from \(\mathcal H_{\ell }([0,2\pi ];\mathbb R^2)\) to itself for every \(\ell >1/2\). Hence, assumption (B) holds for the nonlinear Schrödinger equation (2.16) for any \(I=[0,L]\), \(L\ge 0\) if we set \(\mathcal Y_\ell = \mathcal H_{2\ell +\alpha }([0,2\pi ];\mathbb R^2)\) for \(\alpha >1/2\).

When we equip the nonlinear Schrödinger equation (2.16) with Dirichlet (Neumann) boundary conditions we need to require that \(\ell +\frac{\alpha }{2} \notin \mathbb N_0 + \frac{1}{4}\) (\(\ell +\frac{\alpha }{2} \notin \mathbb N_0 + \frac{3}{4}\)) and, for Dirichlet boundary conditions, we need the potential *V* to be even or satisfy \(V^{(2j+1)}(0)=0\) for \(0\le j< L+\alpha -\frac{1}{4}\). Here \(I = [0,L] {\setminus } (\mathbb N_0 + \frac{1}{4}-\frac{\alpha }{2})\) for Dirichlet boundary conditions and \(I = [0,L] {\setminus } (\mathbb N_0 + \frac{3}{4}-\frac{\alpha }{2})\) for Neumann boundary conditions.

The nonlinearities of the PDEs in the above examples are superposition operators \(f :\mathcal H_{\ell }([0,2\pi ];\mathbb R^d) \rightarrow \mathcal H_{\ell }([0,2\pi ];\mathbb R^d)\) of smooth functions \(f:D \subseteq \mathbb R^d \rightarrow \mathbb R^d\) or restrictions of such operators to spaces encorporating boundary conditions. To prove that these superposition operators satisfy assumption (B) we have employed the following lemma. Part a) of this lemma has already been stated in slightly different form in [7, 14], and parts b) and c) follow from [15].

### **Lemma 2.9**

- (a)
Let \(\rho >0\) and let \(f :\mathcal B_{\mathbb C^d}^\rho \rightarrow \mathbb C^d\) be analytic. If \(\Omega \) is unbounded assume \(f(0)=0\). Then

*f*is also analytic as a function from \(\mathcal B_{\mathcal H_\ell }^R\) to \(\mathcal H_{\ell }:= \mathcal H_{\ell }(\Omega ;\mathbb C^d)\) for every \(\ell > n/2\) and \(R\le \rho /c\) with*c*from (2.19) below. Moreover \(f:\mathcal B_{\mathcal H_\ell }^R \rightarrow \mathcal H_\ell \) and its derivatives up to order*N*are bounded with*N*-dependent bounds for arbitrary \(N\in \mathbb N\). - (b)Let \(f\in \mathcal C_{{\text {b}}}^{N}(D,\mathbb R^d)\) for some open set \(D \subset \mathbb R^d\) and \(N \in \mathbb N\). If \(\Omega \) is unbounded assume \(f(0)=0\). Let \(j \in \mathbb N\) be such that \(j> n/2\). Let \(\mathcal D\) be an \(\mathcal H_j\) bounded subset ofand for \(R>0\), \(k\in \mathbb N\) with \(k\ge j\) let$$\begin{aligned} \{ u \in \mathcal H_j(\Omega ;\mathbb R),~ u(\Omega ) \subset D \} \end{aligned}$$Here \(\mathcal H_k =\mathcal H_{k}(\Omega ;\mathbb R^d) \). Then,$$\begin{aligned} \mathcal D_k = \mathcal D\cap \mathcal B^R_{\mathcal H_k}(0). \end{aligned}$$(2.18)with$$\begin{aligned} f \in \mathcal C_{{\text {b}}}^{N-k}(\mathcal D_k; \mathcal H_k), \quad \text{ for }\quad k \in \{j,\ldots , N\} \end{aligned}$$
*R*-dependent bounds. - (c)Let
*D*,*f*and*j*be as in b) and let \(L> n/2\) be such that \(\lfloor L\rfloor \le N\). Thenwith \(\mathcal D_\ell \) defined as in (2.18).$$\begin{aligned} f \in \mathcal C_{{\text {b}}}^{N-\lfloor \ell \rfloor }(\mathcal D_{\ell }; \mathcal H_{\ell -1}) \quad \text{ for } \text{ all }\quad \ell \in [j,L], \end{aligned}$$

### *Proof*

We restrict to the case \(d=1\). A generalization to \(d>1\) is straightforward.

*f*be analytic on \(\mathcal B_{\mathbb C}^\rho \) and let

*f*around 0 for \(|z|\le \rho \). Let \(g :\mathbb R\rightarrow \mathbb R\) be its majorization

*f*(

*u*), we see that the series converges for every \(u \in \mathcal H_{\ell }\) provided \(\ell > n/2\), and that

*c*is as in (2.19), \(R\le \rho /c\) and \(a_0=0\) if \(\Omega \) is unbounded. In other words,

*f*is analytic and bounded as function from a ball of radius

*R*around 0 in \(\mathcal H_{\ell }= \mathcal H_{\ell }(\Omega ;\mathbb C)\) to \(\mathcal H_{\ell }\). Similarly we see that the same holds for the derivatives of

*f*.

To prove b) note that \(\mathcal D\) is well-defined because by the Sobolev embedding theorem \(\mathcal H_j(\Omega ;\mathbb R) \subseteq \mathcal C_{{\text {b}}}(\Omega ;\mathbb R)\). In [15, Theorem 2.12], the statement was proved in the case \(n=1\). The extension to the case \(n>1\) is straightforward. Here let us just illustrate the idea of the proof for the example \(n=1\), \(N=1\) and \(j=k=1\). Then \(f \in \mathcal C_{{\text {b}}}^1(\mathcal D_1;\mathcal L_2)\) by the Sobolev embedding theorem, but also \(f \in \mathcal C_{{\text {b}}}(\mathcal D_1;\mathcal H_1)\) since for this we only need that \(\partial _x f(u) = f'(u) \partial _x u \in \mathcal L_2\) with uniform bound in \(u \in \mathcal D_1\) which is again true by the Sobolev embedding theorem.

To prove c) note that for \(\ell \in [j,L]\) we know from b) that \(f \in \mathcal C^{N-\lfloor \ell \rfloor }(\mathcal D_{\lfloor \ell \rfloor }; \mathcal H_{\lfloor \ell \rfloor })\). Since \(\mathcal D_{\ell } \subseteq \mathcal D_{\lfloor \ell \rfloor }\) and \(\mathcal H_{\lfloor \ell \rfloor } \subseteq \mathcal H_{ \ell -1}\) this implies \(f \in \mathcal C^{N-\lfloor \ell \rfloor }(\mathcal D_{\ell }; \mathcal H_{\ell -1})\). \(\square \)

## 3 Runge–Kutta time semidiscretizations

In this section we apply an A-stable Runge–Kutta method in time to the evolution equation (1.1), and establish well-posedness and regularity of the numerical method on the infinite dimensional space \(\mathcal Y\).

*s*,

*s*) matrix \(\mathsf a\), and a vector \(\mathsf b\in \mathbb R^s\), we define the corresponding Runge–Kutta method by

*A*to act diagonally on the vector

*W*, i.e., \((AW)^i=AW^i\), and

*A*-stability of the numerical method as follows (cf. [12]):

- (RK1)
\(\mathsf S(z)\) from (3.4) is bounded with \(|\mathsf S(z)|\le 1\) for all \(z\in \mathbb C^-_0\).

- (RK2)
\(\mathsf a\) is invertible and the matrices \({\text {id}}-z\mathsf a\) are invertible for all \(z\in \mathbb C^-_0\).

### *Example 3.1*

Gauss–Legendre collocation methods such the implicit midpoint rule satisfy (RK1) and (RK2) [15, Lemma 3.6].

The following result is needed later on, see also [15, Lemmas 3.10, 3.11, 3.13]:

### **Lemma 3.2**

### *Proof*

Analogously to Theorem 2.2, we require a well-posedness and regularity result for the stage vectors \(W^i\), \(i=1,\ldots , s\), and the numerical method \(\Psi ^h\). The following result is an extension of [15, Theorem 3.14] to non-integer values of \(\ell \).

### **Theorem 3.3**

*W*and numerical method \(\Psi \) which satisfy

*W*depend only on

*R*, (3.5), those afforded by assumption (B) on balls of radius

*R*and on \(\mathsf a\), \(\mathsf b\) as specified by the numerical method.

### *Proof*

*W*as fixed point of the map \(\Pi :\mathcal B^R_{\mathcal Y^s}(0) \times \mathcal B^r_\mathcal Y(0) \times [0,h_*]\rightarrow \mathcal Y^s\), given by

*U*we have

*N*derivatives in

*U*.

This proves statements (3.6a) and also (3.6c) in the case \(k=0\) for *W*. Due to (3.3), these statements also hold true for \(\Psi \). In the case \(k\ne 0\) it follows from the that \(\ell \in I^-\) that the above argument also holds on \(\mathcal Y_{\ell -j}\), \(j=0,\ldots , k\). Hence there is some \(h_*>0\) such that \(W^i, \Psi \in \mathcal C_{{\text {b}}}(\mathcal B^r_{\ell -j}\times [0,h_*];\mathcal B^R_{\ell -j})\), \(j=0,\ldots , k\), \(i=1,\ldots , s\). As shown in [15] for \(U \in \mathcal B^r_{\ell }\) the *h* derivatives up to order *k* can then be obtained by implicit differentiation of \(\Pi (W,U,h)=W(U,h)\) with \(\Pi \) defined above and by differentiating (3.3), cf. the proof of Theorem 2.2. This then implies (3.6c). \(\square \)

*p*if the local error, i.e., the one-step error, of the numerical method is given by the Taylor remainder of order \(p+1\),

*U*(

*t*) satisfies \(U(t) \in \mathcal Y_\ell \) with \(\ell <p+1\), by means of Galerkin truncation.

## 4 Spectral Galerkin truncations

*A*on to the set \({\text {spec}}\ (A)\cap \mathcal B^{m}_{\mathbb C}(0)\), and set \(\mathbb Q_m={\text {id}}-\mathbb P_m\). In this setting we define \(B_m(u_m)=\mathbb P_mB(u_m)\), and consider the projected semilinear evolution equation

*m*.

### **Lemma 4.1**

*R*, \(\omega \) from (2.1), and those afforded by assumption (B) on balls of radius

*R*.

In the case \(B\equiv 0\) it is clear that for \(U^0 \in \mathcal Y_\ell \) we have the estimate \(||\Phi ^t(U^0)-\Phi _m^t(U^0) ||_{\mathcal Y}^{} = \mathcal O(m^{-\ell })\) on any finite interval of existence [0, *T*]. With the presence of a nonlinear perturbation \(B\ne 0\) a similar result can be obtained by a Gronwall type argument as shown in the lemma below, which gives an appropriate bound for the error of the semiflow incurred in Galerkin truncation. Note that similar results for mixed higher order derivatives in time and initial value are obtained, for integer \(\ell \) in [16, Theorems 2.6 and 2.8].

### **Lemma 4.2**

*R*,

*T*, (2.1) and the bounds afforded by (B) on balls of radius \(R+\delta \).

### *Proof*

We also consider an *s*-stage Runge–Kutta method applied to the projected semilinear evolution equation (4.1). We denote by \(w_m=w_m(u^0_m,h)\) the stage vector of this map, and by \(\psi ^h_m(u_m^0)\) the one-step numerical method applied to the projected system (4.1) and define \(W_m=w_m\circ \mathbb P_m\), \(\Psi ^h_m=\psi ^h_m\circ \mathbb P_m\). Similar to Lemma 4.1 and Lemma 4.2, we have the following results regarding the existence, regularity and error under truncation for the projected numerical method. Note that similar results have been obtained, for integer \(\ell \), and mixed derivatives in [16, Theorems 3.2 and 3.6].

### **Lemma 4.3**

*r*is as in (3.6b), with uniform bounds in \(h \in [0,h_*]\), \(m\ge 0\). Furthermore, for \(\ell \in I^-\), \(k\in \mathbb N_0 \), \(k \le \ell \), we have for \(i=1,\ldots , s\),

*R*, (3.5), those afforded by assumption (B) on balls of radius

*R*and on \(\mathsf a\), \(\mathsf b\) as specified by the numerical method.

### *Proof*

## 5 Trajectory error bounds for non-smooth data

*T*], \(0\le nh\le T\), given sufficient regularity of the semiflow and time semidiscretization to bound the local error given by the Taylor expansion to order \(p+1\) as a map

In the rest of this section, equipped with the results of Sect. 4 on the stability of the semiflow and the numerical method under Galerkin, truncation we estimate the growth with *m* of the local error of a Runge–Kutta method (3.1), subject to (RK1) and (RK2), applied to the projected equation (4.1) subject to (A) and (B) for non-smooth initial data. In this setting, by coupling *m* and *h* and balancing the projection error and trajectory error of the projected system, we obtain an estimate for \(q(\ell )\) that describes the convergence of the numerical method for the semilinear evolution equation (1.1) as observed in Fig. 1, see Sect. 5.2.

### 5.1 Preliminaries

We start with some preliminary lemmas.

### **Lemma 5.1**

*m*-dependent bounds for derivatives of \(\Phi _m\)) Assume that the semilinear evolution equation (1.1) satisfies (A) and (B) and choose \(\ell \in I^-\), \(T>0\), \(m_*\ge 0\) and \(R>0\). Then for all \(U^0\) with

*m*-dependent bounds which are uniform in \(U^0\). Moreover for all such \(U^0\), \(\ell \le k \le N\),

*T*,

*R*, (2.1) and the bounds from assumption (B).

### *Proof*

*m*independent bounds.

To prove (5.3c) we proceed by induction over \(k = \lceil \ell \rceil ,\ldots ,N\).

If \(\ell \ge 1\), \(\ell \in \mathbb Z\) then the start of the induction is \(k=\ell \), and the left hand side of (5.3c) is bounded by (5.3b).

*m*in the \(\mathcal Y_{\ell -\lfloor \ell \rfloor }\) norm. Using the Faà di Bruno formula [5] we find that for any \(i \in \mathbb N\), \(i< N\),

*i*replaced by \(\lfloor \ell \rfloor \). Then the second term in the last line of (5.5) is bounded independent of \(m\ge m_*\) due to (5.3b). Furthermore, since \(\partial _t^{\lfloor \ell \rfloor }u_m\in \mathcal Y_{\ell -\lfloor \ell \rfloor }\) by (5.4) with uniform bound in \(m\ge m_*\), we estimate

*k*and assume that (5.3c) holds for all integers

*i*such that \(\ell \le i \le k\). We now use (5.5) with \(i=k\) to estimate \(\Vert \partial _t^{k+1}u_m\Vert _\mathcal Y\). By the first inequality of (2.4) and the induction hypothesis the first term on the second line of (5.5) is \(\mathcal O(m^{k+1-\ell })\). Moreover, by (5.3b) and the induction hypothesis, the \(\mathcal Y\) norm of the second term is of order \(\mathcal O(m^n)\) with \(n=0\) if \(j_{\lceil \ell \rceil } + \cdots + j_k=0\) and

### **Lemma 5.2**

*m*-dependent bounds for derivatives of \(\Psi _m\) and \(W_m\)) Assume that the semilinear evolution equation (1.1) satisfies (A) and (B), and apply a Runge–Kutta method subject to (RK1) and (RK2). Choose \(\ell \in I^-\) and \(k\in \mathbb N_0\) with \(\ell \le k\le N\). Let \(R>0\) and define

*r*as in (3.6b). Then there is \(h_*>0\) such that for \(m\ge 0\) and \(i=1,\ldots , s\),

*m*-dependent bounds which are uniform in \(U \in \mathcal B_\ell ^r \). Moreover

*R*, (3.5), \(\mathsf a\) and \(\mathsf b\) from the numerical method and the bounds afforded by (B) on balls of radius

*R*.

### *Proof*

*n*by \(k-j\)) and (5.8), we can estimate the

*j*-th term in the sum of (5.9) for \(0\le j\le \ell \le k\) as follows:

*j*-th term in the sum of (5.9) for \(j\ge \ell \) as follows:

*k*times in

*h*:

*h*-derivatives of order at most \(k-1\) and are therefore bounded and in particular \(\mathcal O(m^{k-\ell })\), except when \(\beta =j_k=1\) and \(j_\alpha =0\) for \(\alpha \ne k\). So we obtain

### 5.2 Trajectory error for nonsmooth data

Now we are ready to prove our main result:

### **Theorem 5.3**

*R*,

*T*, (2.1), (3.5), \(\mathsf a\), \(\mathsf b\) from the numerical method and the bounds afforded by (B).

*Proof of Theorem* 5.3. The proof consists of several steps, as outlined in the diagram below:

We want to estimate the error of the Runge Kutta time discretization of the evolution equation (first line of the diagram). To do this, in a first step, we discretize in space by a Galerkin truncation. We estimate the projection error and prove regularity of the solution \(u_m(t)\) of the projected system (first column in the diagram). In the second step of the proof we investigate the error of the time discretization of the space-discretized system (third row in the diagram) and couple the spatial discretization parameter *m* with the time step size *h* in suitable way. In the third step of the proof (third column of the diagram) we prove regularity of the space-time discretization and estimate the projection error of the Runge Kutta time discretization. This concludes the proof.

*Step 1*(

*Regularity of solution of the projected system*) In a first step we aim to prove regularity of the continuous solution of the projected system \(u_m(t) = \phi _m^t(\mathbb P_m U^0) =\Phi _m^t( U^0)\) which will be needed later. For the proof we denote

*R*from (5.20) as \(R_\Phi \) to indicate that it is a bound on \(\Phi ^t(U^0)\). We will prove that there is some \(r_\phi >0\) such that

*Step 2*(

*Trajectory error of the time discretized projected system*) Next we aim to estimate the trajectory error of the time discretization of the projected system. First note that by Theorem 4.3 (with

*r*replaced by \(2r_\phi \) and consequently

*R*by \(4 r_\phi \Lambda \)) there is \(h_*>0\) such that for \(m \ge 0\), \(h \in [0,h_*]\) we have \( W^i_h, \Psi ^h_m \in \mathcal C_{{\text {b}}}^1(\mathcal B^{2r_\phi }_0;\mathcal Y) \), \(i=1,\ldots , s\), with uniform bounds in \(m \ge 0\), \(h \in [0,h_*]\). Moreover, using (3.3), (3.5a) and (3.5b) we obtain the following bound for \(h \in [0,h_*]\) to be used later:

*R*replaced by \(r_\phi \). The second supremum in (5.26b) is \(O(m^{p+1-\ell })\) by Lemma 5.2, with \(\mathcal B^r_\ell \) replaced by \(\mathcal B_\ell ^{r_\phi }\) (and

*R*replaced by \(2r_\phi \Lambda \)).

*Step 3*(

*Projection error of numerical trajectory*) We now estimate the global projection error of the numerical method. We will prove that for \(m(h) = h^{-p/(p+1)}\), \(nh\le T\), \(h \in [0,h_*]\),

*r*replaced by \(2r_\psi \) [and consequently

*R*by \(4r_\psi \Lambda \), see (3.6b)] we have

*r*replaced by \(r_\psi \) and

*R*by \(2r_\psi \Lambda \))

### *Example 5.4*

*Cubic nonlinear Schrödinger equation in*\(\mathbb R^3\)) We now consider a cubic nonlinear Schrödinger equation in \(\mathbb R^3\)

*B*(

*U*) is analytic on \(\mathcal Y\) and the same holds true on \(\mathcal Y_\ell = D(A^\ell ) =\mathcal H_{2(\ell +1)}(\mathbb R^3,\mathbb R^2) \) where \(\ell \ge 0\). In this case assumption (B) holds for \(I=[0, L]\) and any \(L>0\). If (5.40) is discretized by the implicit mid point rule and \(U^0 \in \mathcal Y_1=\mathcal H_4\), then from Theorem 5.3 we obtain an order of convergence \(\mathcal O(h^{2/3})\) in the \(\mathcal H_2\)-norm. In [13] a second order Strang type time discretization is used to discretize (5.40) and a better rate of convergence is observed, namely an order of convergence \(\mathcal O(h)\) in the \(\mathcal H_2\)-norm for \(U^0 \in \mathcal H_4\). This is due to the fact that the linear part of the evolution equation (1.1), i.e., \(\dot{U} = AU\), is integrated exactly by this method. We plan to extend the methods of this paper to splitting and exponential integrators in future work.

## Notes

### Acknowledgments

The authors want to thank the Nuffield foundation for the Summer Bursary Scheme under which this project was started in 2009.

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