Efficient Explicit Time Stepping of High Order Discontinuous Galerkin Schemes for Waves

TL;DR

This paper develops efficient, matrix-free algorithms for explicit time stepping in high-order discontinuous Galerkin methods for wave simulations, optimizing performance for complex geometries and higher order accuracy.

Contribution

It introduces a flexible basis change, degree reduction, and performance analysis for ADER and Runge-Kutta schemes, enhancing efficiency and throughput in wave propagation computations.

Findings

ADER outperforms Runge-Kutta at high orders and CFL limits.

Proposed optimizations significantly reduce computational time.

Higher arithmetic intensities improve ADER efficiency.

Abstract

This work presents algorithms for the efficient implementation of discontinuous Galerkin methods with explicit time stepping for acoustic wave propagation on unstructured meshes of quadrilaterals or hexahedra. A crucial step towards efficiency is to evaluate operators in a matrix-free way with sum-factorization kernels. The method allows for general curved geometries and variable coefficients. Temporal discretization is carried out by low-storage explicit Runge-Kutta schemes and the arbitrary derivative (ADER) method. For ADER, we propose a flexible basis change approach that combines cheap face integrals with cell evaluation using collocated nodes and quadrature points. Additionally, a degree reduction for the optimized cell evaluation is presented to decrease the computational cost when evaluating higher order spatial derivatives as required in ADER time stepping. We analyze and compare the performance of state-of-the-art Runge-Kutta schemes and ADER time stepping with the proposed optimizations. ADER involves fewer operations and additionally reaches higher throughput by higher arithmetic intensities and hence decreases the required computational time significantly. Comparison of Runge-Kutta and ADER at their respective CFL stability limit renders ADER especially beneficial for higher orders when the Butcher barrier implies an overproportional amount of stages. Moreover, vector updates in explicit Runge--Kutta schemes are shown to take a substantial amount of the computational time due to their memory intensity.