论文标题
关于高级边界保护方案的最佳细胞平均分解
On Optimal Cell Average Decomposition for High-Order Bound-Preserving Schemes of Hyperbolic Conservation Laws
论文作者
论文摘要
本文介绍了第一个有关寻求最佳细胞平均分解（OCAD）基本问题的系统研究，该研究是由在Zhang-shu框架内构建有效的高阶数值（BP）数值方法引起的。自2010年提议以来，Zhang-Shu框架引起了广泛的关注，并应用于为各种双曲线方程开发许多高阶BP不连续的Galerkin和有限体积方案。框架中的必要成分是数值溶液平均值分解为在某些正交点处的溶液值组合。在过去的十年中，张和Shu最初提出的经典CAD最初被广泛使用。但是，可行的CAD并非唯一，并且不同的CAD会影响理论BP CFL条件，从而影响计算成本。 Zhang和Shu仅检查了1d $ \ Mathbb P^2 $和$ \ Mathbb P^3 $空间，其经典CAD基于高斯 - lobatto quadrature在实现最温和的BP CFL条件的意义上是最佳的。 在本文中，我们建立了一般理论，用于研究1d和2d的笛卡尔网格上的OCAD问题。我们严格地证明，经典CAD对于一般1D $ \ MATHBB P^K $空间和一般2D $ \ Mathbb Q^K $任意$ K $的空间是最佳的。对于广泛使用的2D $ \ Mathbb P^k $空间，经典CAD并不最佳，我们建立了一般的方法来找出真正的OCAD，并提出了更实用的准式CAD，两者都提供了比经典CAD相比提供了温和的BP CFL条件。结果，我们的OCAD和准最佳CAD明显提高了大型双曲线或对流为主方程的高阶BP方案的效率，几乎没有对实现代码进行轻微的局部修改。
This paper presents the first systematic study on the fundamental problem of seeking optimal cell average decomposition (OCAD), which arises from constructing efficient high-order bound-preserving (BP) numerical methods within Zhang--Shu framework. Since proposed in 2010, Zhang--Shu framework has attracted extensive attention and been applied to developing many high-order BP discontinuous Galerkin and finite volume schemes for various hyperbolic equations. An essential ingredient in the framework is the decomposition of the cell averages of the numerical solution into a convex combination of the solution values at certain quadrature points. The classic CAD originally proposed by Zhang and Shu has been widely used in the past decade. However, the feasible CADs are not unique, and different CAD would affect the theoretical BP CFL condition and thus the computational costs. Zhang and Shu only checked, for the 1D $\mathbb P^2$ and $\mathbb P^3$ spaces, that their classic CAD based on the Gauss--Lobatto quadrature is optimal in the sense of achieving the mildest BP CFL conditions. In this paper, we establish the general theory for studying the OCAD problem on Cartesian meshes in 1D and 2D. We rigorously prove that the classic CAD is optimal for general 1D $\mathbb P^k$ spaces and general 2D $\mathbb Q^k$ spaces of arbitrary $k$. For the widely used 2D $\mathbb P^k$ spaces, the classic CAD is not optimal, and we establish the general approach to find out the genuine OCAD and propose a more practical quasi-optimal CAD, both of which provide much milder BP CFL conditions than the classic CAD. As a result, our OCAD and quasi-optimal CAD notably improve the efficiency of high-order BP schemes for a large class of hyperbolic or convection-dominated equations, at little cost of only a slight and local modification to the implementation code.
