Collaborative Research: A Higher Order PDE Toolkit for Computational Mathematics and Astrophysical Turbulence

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Our science and technology-driven society is immensely dependent on robust, reliable and highly accurate simulations of various natural phenomena. The scope of such natural phenomena is vast. In this proposal, the PIs take the task of generating software that can robustly, reliably and with high accuracy simulate phenomena that have to do with the propagation of wave-like motions. Such computational codes are very useful in simulating problems in applied and computational mathematics as well as computational astrophysics. The proposed effort seeks to develop a common software substrate for both those communities. The software is based on advanced computational algorithms and is intended to work very efficiently on modern generations of supercomputers that have been funded by the National Science Foundation. To solve the common problem faced by both communities the PIs develop a toolkit for higher order partial differential equation (PDE) methods. The proposed computational toolkit mainly includes two components. The first one is a multidimensional Riemann solver software that can work with some PDE systems. The second one is Discontinuous Galerkin (DG) schemes for some involution-constrained systems of hyperbolic PDEs such as Euler and magnetohydrodynamics (MHD) flows.