Abstract
In this paper we discuss a new and very efficient implementation of high order accurate arbitrary high order schemes using derivatives 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, and so 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 levels 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.
Highlights
Hyperbolic partial differential equations are omnipresent in the mathematical description of time-dependent processes in fluid and solid mechanics, in engineering and geophysics, as well as in plasma physics, and even in general relativity
In this paper we mainly focus on a rather recent family of schemes, which is of the discontinuous finite element type, namely the so-called discontinuous Galerkin (DG) finite element methods, which were systematically introduced for hyperbolic conservation laws in a well-known series of papers by Cockburn and Shu and collaborators [27,28,29,30,31]
Ω = [0, 2π ]3 up to t = 1 and carried out with high order ADER-DG schemes in Reference [57] is reported in the Table 2 below, where we show a direct comparison with high order Runge-Kutta discontinuous Galerkin schemes
Summary
Hyperbolic partial differential equations are omnipresent in the mathematical description of time-dependent processes in fluid and solid mechanics, in engineering and geophysics, as well as in plasma physics, and even in general relativity. The main challenge of nonlinear hyperbolic PDE arises from the fact that they can contain at the same time smooth solutions (like sound waves) as well as small scale structures (e.g., turbulent vortices), but they can develop discontinuous solutions (shock waves) after finite time, even when starting from perfectly smooth initial data. In the 1940ies and 1950ies, major steps forward in numerical methods for hyperbolic PDE have been made in the ground-breaking contributions of von Neumann and Richtmyer [20] and Godunov [21] While the former introduce an artificial viscosity to stabilize the numerical scheme in the presence of discontinuities, the latter constructs his scheme starting from the most elementary problem in hyperbolic conservation laws for which an exact solution is still available, the so-called Riemann problem. These properties make the method well suitable for high performance computing (HPC)
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