A Block Krylov Subspace Implementation of the Time-Parallel Paraexp Method and its Extension for Nonlinear Partial Differential Equations

TL;DR

This paper introduces a parallel time integration method for nonlinear PDEs using a block Krylov subspace implementation of Paraexp, enabling efficient and scalable solutions for complex equations like advection-diffusion and Burgers.

Contribution

It extends the Paraexp method with a block Krylov approach and waveform relaxation for nonlinear PDEs, improving parallel efficiency and scalability.

Findings

Excellent parallel scaling for linear problems

Good scaling for nonlinear Burgers equation

Efficient decoupling of subproblems via superposition

Abstract

A parallel time integration method for nonlinear partial differential equations is proposed. It is based on a new implementation of the Paraexp method for linear partial differential equations (PDEs) employing a block Krylov subspace method. For nonlinear PDEs the algorithm is based on our Paraexp implementation within a waveform relaxation. The initial value problem is solved iteratively on a complete time interval. Nonlinear terms are treated as a source term, provided by the solution from the previous iteration. At each iteration, the problem is decoupled into independent subproblems by the principle of superposition. The decoupled subproblems are solved fast by exponential integration, based on a block Krylov method. The new time integration is demonstrated for the one-dimensional advection-diffusion equation and the viscous Burgers equation. Numerical experiments confirm excellent parallel scaling for the linear advection-diffusion problem, and good scaling in case the nonlinear Burgers equation is simulated.