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Abstract We present a new numerical model for solving the Chew–Goldberger–Low system of equations describing a bi-Maxwellian plasma in a magnetic field. Heliospheric and geospace environments are often observed to be in an anisotropic state with distinctly different parallel and perpendicular pressure components. The Chew–Goldberger–Low (CGL) system represents the simplest leading order correction to the common isotropic MHD model that still allows the incorporation of the latter’s most desirable features. However, the CGL system presents several numerical challenges: the system is not in conservation form, the source terms are stiff, and unlike MHD, it is prone to a loss of hyperbolicity if the parallel and perpendicular pressures become too different. The usual cure is to bring the parallel and perpendicular pressures closer to one another, but that has usually been done in an ad hoc manner. We present a physics-informed method of pressure relaxation based on the idea of pitch-angle scattering that keeps the numerical system hyperbolic and naturally leads to zero anisotropy in the limit of very large plasma beta. Numerical codes based on the CGL equations can, therefore, be made to function robustly for any magnetic field strength, including the limit where the magnetic field approaches zero. The capabilities of our new algorithm are demonstrated using several stringent test problems that provide a comparison of the CGL equations in the weakly and strongly collisional limits. This includes a test problem that mimics the interaction of a shock with a magnetospheric environment in 2D. 

GPU computing is expected to play an integral part in all modern Exascale supercomputers. It is also expected that higher order Godunov schemes will make up about a significant fraction of the application mix on such supercomputers. It is, therefore, very important to prepare the community of users of higher order schemes for hyperbolic PDEs for this emerging opportunity. Not every algorithm that is used in the space-time update of the solution of hyperbolic PDEs will take well to GPUs. However, we identify a small core of algorithms that take exceptionally well to GPU computing. Based on an analysis of available options, we have been able to identify weighted essentially non-oscillatory (WENO) algorithms for spatial reconstruction along with arbitrary derivative (ADER) algorithms for time extension followed by a corrector step as the winning three-part algorithmic combination. Even when a winning subset of algorithms has been identified, it is not clear that they will port seamlessly to GPUs. The low data throughput between CPU and GPU, as well as the very small cache sizes on modern GPUs, implies that we have to think through all aspects of the task of porting an application to GPUs. For that reason, this paper identifies the techniques and tricks needed for making a successful port of this very useful class of higher order algorithms to GPUs. Application codes face a further challenge—the GPU results need to be practically indistinguishable from the CPU results—in order for the legacy knowledge bases embedded in these applications codes to be preserved during the port of GPUs. This requirement often makes a complete code rewrite impossible. For that reason, it is safest to use an approach based on OpenACC directives, so that most of the code remains intact (as long as it was originally well-written). This paper is intended to be a one-stop shop for anyone seeking to make an OpenACC-based port of a higher order Godunov scheme to GPUs. We focus on three broad and high-impact areas where higher order Godunov schemes are used. The first area is computational fluid dynamics (CFD). The second is computational magnetohydrodynamics (MHD) which has an involution constraint that has to be mimetically preserved. The third is computational electrodynamics (CED) which has involution constraints and also extremely stiff source terms. Together, these three diverse uses of higher order Godunov methodology, cover many of the most important applications areas. In all three cases, we show that the optimal use of algorithms, techniques, and tricks, along with the use of OpenACC, yields superlative speedups on GPUs. As a bonus, we find a most remarkable and desirable result: some higher order schemes, with their larger operations count per zone, show better speedup than lower order schemes on GPUs. In other words, the GPU is an optimal stratagem for overcoming the higher computational complexities of higher order schemes. Several avenues for future improvement have also been identified. A scalability study is presented for a real-world application using GPUs and comparable numbers of high-end multicore CPUs. It is found that GPUs offer a substantial performance benefit over comparable number of CPUs, especially when all the methods designed in this paper are used.