Multigrid reduction in time for non-linear hyperbolic equations

Time-parallel algorithms seek greater concurrency by decomposing the temporal domain of a partial differential equation, providing possibilities for accelerating the computation of its solution. While parallelisation in time has allowed remarkable speed-ups in applications involving parabolic equations, its effectiveness in the hyperbolic framework remains debatable: growth of instabilities and slow convergence are both strong issues in this case, which often negate most of the advantages from time-parallelisation. Here, we focus on the Multigrid Reduction in Time algorithm, investigating in detail its performance when applied to non-linear conservation laws with a variety of discretisation schemes. Specific attention is given to high-accuracy Weighted Essentially Non-Oscillatory reconstructions, coupled with Strong Stability Preserving integrators, which are often the discretisations of choice for such equations. A technique to improve the performance of MGRIT when applied to a low-order, more dissipative scheme is also outlined. This study aims at identifying the main causes for degradation in the convergence speed of the algorithm and finds the Courant-Friedrichs-Lewy limit to be the principal determining factor. Key words. parallel-in-time integration, multigrid, conservation laws, WENO, high-order methods. AMS subject classifications. 65M08, 35L65, 65M55, 65Y05, 65Y20