Parallel-in-time solution of hyperbolic PDE systems via characteristic-variable block preconditioning

TL;DR

This paper introduces a parallel-in-time method for hyperbolic PDE systems using characteristic-variable block preconditioning, enabling efficient solution of both linear and nonlinear equations with weak inter-variable coupling.

Contribution

The paper develops a novel block preconditioning approach in characteristic variables for parallel-in-time solutions of hyperbolic PDEs, applicable to linear and nonlinear systems.

Findings

Effective parallel-in-time solution for hyperbolic PDEs demonstrated

Preconditioning in characteristic variables reduces inter-variable coupling

Numerical results show efficiency for acoustics, shallow water, and Euler equations

Abstract

We consider the parallel-in-time solution of hyperbolic partial differential equation (PDE) systems in one spatial dimension, both linear and nonlinear. In the nonlinear setting, the discretized equations are solved with a preconditioned residual iteration based on a global linearization. The linear(ized) equation systems are approximately solved parallel-in-time using a block preconditioner applied in the characteristic variables of the underlying linear(ized) hyperbolic PDE. This change of variables is motivated by the observation that inter-variable coupling for characteristic variables is weak relative to intra-variable coupling, at least locally where spatio-temporal variations in the eigenvectors of the associated flux Jacobian are sufficiently small. For an $\ell$-dimensional system of PDEs, applying the preconditioner consists of solving a sequence of $\ell$ scalar linear(ized)-advection-like problems, each being associated with a different characteristic wave-speed in the underlying linear(ized) PDE. We approximately solve these linear advection problems using multigrid reduction-in-time (MGRIT); however, any other suitable parallel-in-time method could be used. Numerical examples are shown for the (linear) acoustics equations in heterogeneous media, and for the (nonlinear) shallow water equations and Euler equations of gas dynamics with shocks and rarefactions.