Parallel-in-Time High-Order Multiderivative IMEX Solvers

Abstract

In this work, we present a novel class of high-order time integrators for the numerical solution of ordinary differential equations. These integrators are of the two-derivative type, i.e., they take into account not only the first, but also the second temporal derivative of the unknown solution. While being motivated by two-derivative Runge–Kutta schemes, the methods themselves are of the predictor-corrector type, constructed in such a way that time-parallelism through pipelining is possible. This means that predictor and corrector steps can be computed simultaneously on different processors. Two variants are shown, the second-one being of Gauss–Seidel-type and hence having lower storage requirements. It turns out that this second variant is not only low-storage, but also performs better in numerical simulations. The algorithms presented can be cast as implicit two-step-two-derivative-multi-stage methods for which we present a detailed mathematical convergence analysis. Subsequently, numerical results are shown, demonstrating first the algorithms’ ability to cope with very stiff equations and showing up to eighth order of accuracy. Following, the parallel-in-time capabilities are illustrated. We conclude that the class of methods is very well-suited if the time parallelization of very stiff equations is an option.

Matrices of “One-Step Equivalent”

In the following, the matrices required for the formulation of the “one-step equivalent” of Algorithm 1 in Eq. (9) are given. We start by defining the matrices ${\mathfrak {I}},~{\mathfrak {C}}^{(1)},~{\mathfrak {C}}^{(2)}\in \mathbb {R}^{s\times s}$ with

$$\begin{aligned} {\mathfrak {I}}:=\begin{pmatrix} 0 &{} 0 &{} 0 &{} \cdots &{} 0\\ 0 &{} 1 &{} 0 &{} \cdots &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} 1 \end{pmatrix},~ {\mathfrak {C}}^{(1)}:=\begin{pmatrix} 0&{}0&{}\cdots &{}0\\ c_2&{}0&{}\cdots &{}0\\ \vdots &{}\vdots &{}\ddots &{}\vdots \\ c_l&{}0&{}\cdots &{}0 \end{pmatrix},~\text {and}~ {\mathfrak {C}}^{(2)}:=\begin{pmatrix} 0&{}0&{}\cdots &{}0\\ c_2^2&{}0&{}\cdots &{}0\\ \vdots &{}\vdots &{}\ddots &{}\vdots \\ c_l^2&{}0&{}\cdots &{}0 \end{pmatrix}. \end{aligned}$$

With this, the matrices $B_I$ and $\tilde{B}_I$ for the stiff contribution are given by

$$\begin{aligned} \begin{aligned} B_I&= \begin{pmatrix} {{\,\mathrm{diag}\,}}(c) &{} 0 &{} 0 &{} \cdots &{} \cdots &{} 0\\ B^{(1)}-{\mathfrak {I}} &{} {\mathfrak {I}} &{} 0 &{} \cdots &{} \cdots &{} 0\\ 0 &{} B^{(1)}-{\mathfrak {I}} &{} {\mathfrak {I}} &{} \cdots &{} \cdots &{} 0\\ \vdots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} \vdots \\ 0 &{} \cdots &{} \cdots &{} \cdots &{} B^{(1)}-{\mathfrak {I}} &{} {\mathfrak {I}}\\ \end{pmatrix},\\ \tilde{B}_I&= \begin{pmatrix} -{{\,\mathrm{diag}\,}}(c^2) &{} 0 &{} 0 &{} \cdots &{} \cdots &{} 0\\ 2B^{(2)}+{\mathfrak {I}} &{} -{\mathfrak {I}} &{} 0 &{} \cdots &{} \cdots &{} 0\\ 0 &{} 2B^{(2)}+{\mathfrak {I}} &{} -{\mathfrak {I}} &{} \cdots &{} \cdots &{} 0\\ \vdots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} \vdots \\ 0 &{} \cdots &{} \cdots &{} \cdots &{} 2B^{(2)}+{\mathfrak {I}} &{} -{\mathfrak {I}}\\ \end{pmatrix}. \end{aligned} \end{aligned}$$

Note that c, $B^{(1)}$ and $B^{(2)}$ are defined by the chosen Hermite–Birkhoff quadrature rule, see Eqs. (2)–(4). The matrices $B_E$ and $\tilde{B}_E$ of the non-stiff contribution can be found as

$$\begin{aligned} \begin{aligned} B_E = \begin{pmatrix} {\mathfrak {C}}^{(1)} &{} 0 &{} 0 &{} \cdots &{} \cdots &{} 0\\ B^{(1)}&{} 0 &{} 0 &{} \cdots &{} \cdots &{} 0\\ 0 &{} B^{(1)} &{} 0 &{} \cdots &{} \cdots &{} 0\\ \vdots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} \vdots \\ 0 &{} \cdots &{} \cdots &{} \cdots &{} B^{(1)} &{} 0\\ \end{pmatrix},\quad \tilde{B}_E = \begin{pmatrix} {\mathfrak {C}}^{(2)} &{} 0 &{} 0 &{} \cdots &{} \cdots &{} 0\\ 2B^{(2)} &{} 0 &{} 0 &{} \cdots &{} \cdots &{} 0\\ 0 &{} 2B^{(2)} &{} 0 &{} \cdots &{} \cdots &{} 0\\ \vdots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} \vdots \\ 0 &{} \cdots &{} \cdots &{} \cdots &{} 2B^{(2)} &{} 0\\ \end{pmatrix}. \end{aligned} \end{aligned}$$