Parallel-in-time solution of scalar nonlinear conservation laws

TL;DR

This paper presents a parallel-in-time method for solving scalar nonlinear conservation laws using a combination of finite-volume spatial discretization, high-order Runge-Kutta time integration, and a novel multigrid reduction-in-time approach that efficiently handles shocks and rarefactions.

Contribution

It introduces a new parallel-in-time solution framework with a novel coarse-grid operator for nonlinear conservation laws, extending previous linear hyperbolic problem methods.

Findings

Solver converges in few iterations regardless of mesh size

Effective handling of shocks and rarefactions in numerical tests

Convergence rate independent of mesh resolution

Abstract

We consider the parallel-in-time solution of scalar nonlinear conservation laws in one spatial dimension. The equations are discretized in space with a conservative finite-volume method using weighted essentially non-oscillatory (WENO) reconstructions, and in time with high-order explicit Runge-Kutta methods. The solution of the global, discretized space-time problem is sought via a nonlinear iteration that uses a novel linearization strategy in cases of non-differentiable equations. Under certain choices of discretization and algorithmic parameters, the nonlinear iteration coincides with Newton's method, although, more generally, it is a preconditioned residual correction scheme. At each nonlinear iteration, the linearized problem takes the form of a certain discretization of a linear conservation law over the space-time domain in question. An approximate parallel-in-time solution of the linearized problem is computed with a single multigrid reduction-in-time (MGRIT) iteration, however, any other effective parallel-in-time method could be used in its place. The MGRIT iteration employs a novel coarse-grid operator that is a modified conservative semi-Lagrangian discretization and generalizes those we have developed previously for non-conservative scalar linear hyperbolic problems. Numerical tests are performed for the inviscid Burgers and Buckley--Leverett equations. For many test problems, the solver converges in just a handful of iterations with convergence rate independent of mesh resolution, including problems with (interacting) shocks and rarefactions.