论文标题
经典的凸式分解是在多个维度上用于限制性保护方案的最佳选择吗？
Is the Classic Convex Decomposition Optimal for Bound-Preserving Schemes in Multiple Dimensions?
论文作者
论文摘要
自从[X.张和C.-W。 Shu，J。Comput。 Phys。，229：3091--3120，2010]，Zhang-shu框架吸引了广泛的关注，并激发了许多绑定（BP）高阶不连续的Galerkin和有限的体积方案。框架中的一个关键成分是将数值溶液的单元平均值分解为某些正交点处的溶液值组合，这有助于将高阶方案作为正式一阶方案的凸组组合重写。在过去的十年中，张和Shu最初提出的经典凸分解最初被广泛使用。仅针对1D二次和立方多项式空间进行了验证，即在达到最轻微的BP CFL条件的意义上，经典分解是最佳的。但是，尚不清楚经典分解在多个维度上是否最佳。在本文中，我们发现经典的多维分解基于高斯（Lobatto）和高斯四倍体的张量产物通常不是最佳的，并且我们发现了分别在2和3D多发射空间的全新替代分解，分别是2和3的多发空间，高达2和3。我们的新分解允许比经典的分解时间更大，此外，严格证明它是最佳的BP CFL条件是最佳的，但所需的节点较少。这种最佳凸的分解的发现是高度不平凡但有意义的，因为它可能会改善大型双曲线或对流为主导的方程式的高阶BP方案，而仅需对实现代码进行轻微的局部修改。提供了几个数值示例，以进一步验证我们在效率方面使用我们最佳分解的优势。
Since proposed in [X. Zhang and C.-W. Shu, J. Comput. Phys., 229: 3091--3120, 2010], the Zhang--Shu framework has attracted extensive attention and motivated many bound-preserving (BP) high-order discontinuous Galerkin and finite volume schemes for various hyperbolic equations. A key 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, which helps to rewrite high-order schemes as convex combinations of formally first-order schemes. The classic convex decomposition originally proposed by Zhang and Shu has been widely used over the past decade. It was verified, only for the 1D quadratic and cubic polynomial spaces, that the classic decomposition is optimal in the sense of achieving the mildest BP CFL condition. Yet, it remained unclear whether the classic decomposition is optimal in multiple dimensions. In this paper, we find that the classic multidimensional decomposition based on the tensor product of Gauss--Lobatto and Gauss quadratures is generally not optimal, and we discover a novel alternative decomposition for the 2D and 3D polynomial spaces of total degree up to 2 and 3, respectively, on Cartesian meshes. Our new decomposition allows a larger BP time step-size than the classic one, and moreover, it is rigorously proved to be optimal to attain the mildest BP CFL condition, yet requires much fewer nodes. The discovery of such an optimal convex decomposition is highly nontrivial yet meaningful, as it may lead to an improvement of high-order BP schemes for a large class of hyperbolic or convection-dominated equations, at the cost of only a slight and local modification to the implementation code. Several numerical examples are provided to further validate the advantages of using our optimal decomposition over the classic one in terms of efficiency.
