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.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 also develop discontinuous solutions (shock waves) after finite times, even when starting from perfectly smooth initial data. These discontinuities were first discovered by Bernhard Riemann in his ground breaking work on the propagation of waves of finite amplitude in air [1,2], where the term finite should actually be understood in the sense of large, rather than simple sound waves of infinitesimal strength that have been considered in the times before Riemann. In the abstract of his work, Riemann stated that his discovery of the shock waves might probably not be of practical use for applied and experimental science, but should be mainly understood as a contribution to the theory of nonlinear partial differential equations. Several decades later, shock waves were also observed experimentally, thus confirming the new and groundbreaking mathematical concept of Riemann.The connection between symmetries and conservation laws were established in the work of Emmy Noether [3] at the beginning of the 20th century, while the first methods for the numerical solution of hyperbolic conservation laws go back to famous mathematicians such as . The connection between hyperbolic conservation laws, symmetric hyperbolic systems in the sense of Friedrichs [8] and thermodynamics was established for the first time by Godunov in 1961 [9] and was rediscovered again by Friedrichs and Lax in 1971 [10]. Within this theoretical framework of symmetric hyperbolic and thermodynamically compatible (SHTC) systems, established by Godunov and Romenski [11,12] it is possible to write down the Euler equ...

ExaHyPE: An engine for parallel dynamically adaptive simulations of wave problems

“…This implies that the PDE terms are computed through SIMD operations instead of scalar ones and requires some work from the user. This has been found to greatly improve performance [50]. Instead of the standard Picard iteration, we provide a continuous extension Runge-Kutta scheme (CERK) to yield an initial guess to the Picard iteration, leading to a much lower number of required iterations.…”

A new continuum model for general relativistic viscous heat-conducting media

“…5 for a rather general set of initial conditions containing even discontinuities with the ADER family of methods designed specifically for hyperbolic PDEs, e.g. see [16,37,38], suggests that the system (40) is likely hyperbolic at least in a certain vicinity of equilibrium.…”

“…Finally, we presented a 3+1 split of the governing equations in Sec. 5.1 which are then solved using the ADER-DG family of high-order numerical schemes [14,15,16,38,37] designed specifically for hyperbolic partial differential equations, see Sec. 5.…”

Advanced Numerical Methods in Applied Sciences

“…Arbitrarily high-order schemes using derivatives discontinuous Galerkin (ADER-DG) finite element methods are studied in [10]. The proposed methods are applicable to a wide class of nonlinear systems of partial differential equations, and are aimed at efficiently scaling on massively parallel supercomputers, as is testified by the numerical tests.…”