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

我们的科学和技术驱动的社会极大地依赖于对各种自然现象的强大，可靠和高度准确的模拟。这种自然现象的范围是广阔的。在这个提议中，PI承担了生成软件的任务，这些软件可以鲁棒、可靠和高精度地模拟与波动传播有关的现象。这种计算代码在模拟应用数学和计算数学以及计算天体物理学中的问题时非常有用。 拟议的努力旨在为这两个社区开发一个共同的软件基础。该软件基于先进的计算算法，旨在由美国国家科学基金会资助的现代超级计算机上非常有效地工作。为了解决两个社区面临的共同问题，PI开发了高阶偏微分方程（PDE）方法的工具包。建议的计算工具包主要包括两个组件。第一个是一个多维黎曼求解器软件，可以与一些PDE系统。第二类是离散Galerkin（DG）格式，用于求解对合约束的双曲型偏微分方程组，如Euler方程和磁流体力学（MHD）方程。

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.