双曲型守恒律高阶保界格式的最优胞平均分解
On Optimal Cell Average Decomposition for High-Order Bound-Preserving Schemes of Hyperbolic Conservation Laws
摘要：
本文首次系统地研究了在Zhang-Shu框架下构造高效高阶保界数值方法的基本问题--最优胞平均分解（OCAD）。Zhang-Shu框架自2010年提出以来，引起了广泛的关注，并被应用于发展了许多高阶BP间断Galerkin格式和各种双曲型方程的有限体积格式。在框架中的一个重要组成部分是分解的细胞平均值的数值解到一个凸组合的解决方案的价值在某些正交点。最初由张和舒提出的经典CAD在过去的十年中得到了广泛的应用。然而，可行的CAD并不是唯一的，不同的CAD会影响理论BP CFL条件，从而影响计算成本。Zhang和Shu只检查了一维$\mathbb P^2$和$\mathbb P^3$空间，他们基于Gauss-Lobatto求积的经典CAD在实现最温和的BP CFL条件的意义上是最优的。本文建立了研究一维和二维笛卡尔网格上OCAD问题的一般理论。严格证明了经典CAD对一般一维$\mathbb P^k$空间和任意$k$的一般二维$\mathbb Q^k$空间都是最优的.对于广泛使用的二维$\mathbb P^k$空间，经典CAD不是最优的，我们建立了寻找真正OCAD的一般方法，并提出了一种更实用的准最优CAD，这两种方法都提供了比经典CAD更温和的BP CFL条件。因此，我们的OCAD和准最优CAD显着提高了效率的高阶BP计划的一大类双曲或对流占优的方程，在很小的成本，只有轻微的和局部的修改的实施代码。
英文摘要：
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.
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