Fast multigrid reduction-in-time for advection via modified semi-Lagrangian coarse-grid operators

TL;DR

This paper introduces a novel semi-Lagrangian-based coarse-grid operator for multigrid reduction-in-time methods, enabling fast, scalable, and efficient parallel time integration of linear advection PDEs, outperforming previous approaches.

Contribution

It presents a new semi-Lagrangian coarse-grid operator that approximates the ideal operator, significantly improving MGRIT efficiency for advection-dominated PDEs.

Findings

Substantial speed-ups over sequential methods in 1D and 2D advection problems.

Effective for high-order discretizations up to order five.

First practical method with scalable iteration counts for advection in parallel-in-time algorithms.

Abstract

Many iterative parallel-in-time algorithms have been shown to be highly efficient for diffusion-dominated partial differential equations (PDEs), but are inefficient or even divergent when applied to advection-dominated PDEs. We consider the application of the multigrid reduction-in-time (MGRIT) algorithm to linear advection PDEs. The key to efficient time integration with this method is using a coarse-grid operator that provides a sufficiently accurate approximation to the the so-called ideal coarse-grid operator. For certain classes of semi-Lagrangian discretizations, we present a novel semi-Lagrangian-based coarse-grid operator that leads to fast and scalable multilevel time integration of linear advection PDEs. The coarse-grid operator is composed of a semi-Lagrangian discretization followed by a correction term, with the correction designed so that the leading-order truncation error of the composite operator is approximately equal to that of the ideal coarse-grid operator. Parallel results show substantial speed-ups over sequential time integration for variable-wave-speed advection problems in one and two spatial dimensions, and using high-order discretizations up to order five. The proposed approach establishes the first practical method that provides small and scalable MGRIT iteration counts for advection problems.