Balsara Dinshaw S, Käppeli Roger
Physics and ACMS Departments, University of Notre Dame, Notre Dame, IN USA.
Seminar for Applied Mathematics (SAM), Department of Mathematics, ETH Zürich, CH-8092 Zurich, Switzerland.
Commun Appl Math Comput. 2022;4(3):945-985. doi: 10.1007/s42967-021-00166-x. Epub 2022 Jan 19.
This paper examines a class of involution-constrained PDEs where some part of the PDE system evolves a vector field whose curl remains zero or grows in proportion to specified source terms. Such PDEs are referred to as curl-free or curl-preserving, respectively. They arise very frequently in equations for hyperelasticity and compressible multiphase flow, in certain formulations of general relativity and in the numerical solution of Schrödinger's equation. Experience has shown that if nothing special is done to account for the curl-preserving vector field, it can blow up in a finite amount of simulation time. In this paper, we catalogue a class of DG-like schemes for such PDEs. To retain the globally curl-free or curl-preserving constraints, the components of the vector field, as well as their higher moments, must be collocated at the edges of the mesh. They are updated using potentials collocated at the vertices of the mesh. The resulting schemes: (i) do not blow up even after very long integration times, (ii) do not need any special cleaning treatment, (iii) can operate with large explicit timesteps, (iv) do not require the solution of an elliptic system and (v) can be extended to higher orders using DG-like methods. The methods rely on a special curl-preserving reconstruction and they also rely on multidimensional upwinding. The Galerkin projection, highly crucial to the design of a DG method, is now conducted at the edges of the mesh and yields a weak form update that uses potentials obtained at the vertices of the mesh with the help of a multidimensional Riemann solver. A von Neumann stability analysis of the curl-preserving methods is conducted and the limiting CFL numbers of this entire family of methods are catalogued in this work. The stability analysis confirms that with the increasing order of accuracy, our novel curl-free methods have superlative phase accuracy while substantially reducing dissipation. We also show that PP-like methods, which only evolve the lower moments while reconstructing the higher moments, retain much of the excellent wave propagation characteristics of the DG-like methods while offering a much larger CFL number and lower computational complexity. The quadratic energy preservation of these methods is also shown to be excellent, especially at higher orders. The methods are also shown to be curl-preserving over long integration times.
本文研究了一类对合约束的偏微分方程（PDE），其中PDE系统的某些部分演化出一个向量场，该向量场的旋度保持为零或与特定源项成比例增长。这类PDE分别被称为无旋或保旋的。它们在超弹性和可压缩多相流方程、广义相对论的某些表述以及薛定谔方程的数值解中经常出现。经验表明，如果不对保旋向量场进行特殊处理，它可能在有限的模拟时间内爆炸。在本文中，我们为这类PDE编目了一类类似间断伽辽金（DG）的格式。为了保持全局无旋或保旋约束，向量场的分量及其高阶矩必须配置在网格的边缘。它们使用配置在网格顶点的势进行更新。由此产生的格式：（i）即使在很长的积分时间后也不会爆炸，（ii）不需要任何特殊的清理处理，（iii）可以使用大的显式时间步长运行，（iv）不需要求解椭圆系统，（v）可以使用类似DG的方法扩展到更高阶。这些方法依赖于一种特殊的保旋重构，并且还依赖于多维迎风。对DG方法设计至关重要的伽辽金投影现在在网格的边缘进行，并产生一个弱形式更新，该更新使用借助多维黎曼求解器在网格顶点获得的势。对保旋方法进行了冯·诺依曼稳定性分析，并在本文中编目了这整个方法族的极限库朗数（CFL数）。稳定性分析证实，随着精度阶数的增加，我们新颖的无旋方法具有卓越的相位精度，同时大幅降低耗散。我们还表明，类似PP的方法，即在重构高阶矩时只演化低阶矩，在提供大得多的CFL数和更低的计算复杂度的同时，保留了类似DG方法的许多出色的波传播特性。这些方法的二次能量守恒也被证明是出色的，特别是在高阶时。这些方法还被证明在长时间积分中是保旋的。
