Efficient implementation of ADER discontinuous Galerkin schemes for a scalable hyperbolic PDE engine

TL;DR

This paper presents a highly efficient implementation of ADER-DG schemes optimized for modern supercomputers, enabling large-scale simulations of complex hyperbolic PDE systems with excellent scalability and performance.

Contribution

The paper introduces a scalable, communication-avoiding ADER-DG implementation suitable for exascale computing, demonstrating large-scale hyperbolic PDE simulations.

Findings

Achieved strong scaling up to 180,000 CPU cores.

Performed the largest high-order ADER-DG runs for nonlinear hyperbolic PDEs.

Provided performance comparison with Runge-Kutta DG schemes.

Abstract

In this paper we discuss a new and very efficient implementation of high order accurate ADER discontinuous Galerkin (ADER-DG) finite element schemes on modern massively parallel supercomputers. The numerical methods apply to a very broad class of nonlinear systems of hyperbolic partial differential equations. ADER-DG schemes are by construction communication avoiding and cache blocking and are furthermore very well-suited for vectorization, so that they appear to be a good candidate for the future generation of exascale supercomputers. We introduce the numerical algorithm and show some applications to a set of hyperbolic equations with increasing level of complexity, ranging from the compressible Euler equations over the equations of linear elasticity and the unified Godunov-Peshkov-Romenski (GPR) model of continuum mechanics to general relativistic magnetohydrodynamics (GRMHD) and the Einstein field equations of general relativity. We present strong scaling results of the new ADER-DG schemes up to 180,000 CPU cores. To our knowledge, these are the largest runs ever carried out with high order ADER-DG schemes for nonlinear hyperbolic PDE systems. We also provide a detailed performance comparison with traditional Runge-Kutta DG schemes.