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

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.

26 pages, 10 figures, 2 tables

References in corpus (10)

• Simulating Magnetohydrodynamical Flow with Constrained Transport and Adaptive Mesh Refinement; Algorithms & Tests of the AstroBEAR Code
• Central WENO schemes for hyperbolic conservation laws on fixed and moving unstructured meshes
• Arbitrary-Lagrangian-Eulerian discontinuous Galerkin schemes with a posteriori subcell finite volume limiting on moving unstructured meshes
• ADER discontinuous Galerkin schemes for general-relativistic ideal magnetohydrodynamics
• Space-time adaptive ADER-DG schemes for dissipative flows: compressible Navier-Stokes and resistive MHD equations
• Conformal and covariant Z4 formulation of the Einstein equations: strongly hyperbolic first-order reduction and solution with discontinuous Galerkin schemes
• Well balanced Arbitrary-Lagrangian-Eulerian finite volume schemes on moving nonconforming meshes for the Euler equations of gasdynamics with gravity
• A simple diffuse interface approach on adaptive Cartesian grids for the linear elastic wave equations with complex topography
• On the Eigenvalues of the ADER-WENO Galerkin Predictor
• Stop talking to me -- a communication-avoiding ADER-DG realisation