论文标题
高维双曲线PDE的第五阶WENO方案的快速稀疏网格模拟
Fast sparse grid simulations of fifth order WENO scheme for high dimensional hyperbolic PDEs
论文作者
论文摘要
加权本质上是非振荡（WENO）方案是一种流行的高阶准确数值方法，用于求解双曲线偏微分方程（PDES）。但是，当空间尺寸较高时，空间网格点的数量会大大增加。它通过使用非线性高阶精度WENO方案（例如第五阶WENO方案），导致数值模拟中的大量操作和计算成本。如何通过高级WENO方法实现高空间维度PDE的快速模拟是一个具有挑战性且重要的问题。在文献中，稀疏网格技术已被开发为用于高维问题的非常有效的近似工具。在最近的工作[Lu，Chen和Zhang，纯和应用数学季刊，14（2018）57-86]中，设计了具有稀疏网格组合技术的三阶有限差异方法，旨在求解多维超值双重方程，包括线性对流方程和非线性汉堡方程。在应用程序问题中，通常首选高于三阶的WENO方案，以有效地解决复杂的解决方案结构。在本文中，我们将方法扩展到高阶WENO模拟，特别是第五阶WENO方案。第五阶Weno插值应用于稀疏网格组合技术的延长部分，以处理不连续的溶液。首先解决了基准问题，以证明保存了大量的CPU时间，而WENO方案的第五阶准确度和稳定性都可以保留用于稀疏网格上的模拟。然后将第五阶稀疏网格WENO方法应用于由高维弗拉索夫（Vlasov）建模的动力学问题，通过与常规单网格上的仿真进行比较，进一步证明了计算成本的大量节省。
The weighted essentially non-oscillatory (WENO) schemes are a popular class of high order accurate numerical methods for solving hyperbolic partial differential equations (PDEs). However when the spatial dimensions are high, the number of spatial grid points increases significantly. It leads to large amount of operations and computational costs in the numerical simulations by using nonlinear high order accuracy WENO schemes such as a fifth order WENO scheme. How to achieve fast simulations by high order WENO methods for high spatial dimension hyperbolic PDEs is a challenging and important question. In the literature, sparse-grid technique has been developed as a very efficient approximation tool for high dimensional problems. In a recent work [Lu, Chen and Zhang, Pure and Applied Mathematics Quarterly, 14 (2018) 57-86], a third order finite difference WENO method with sparse-grid combination technique was designed to solve multidimensional hyperbolic equations including both linear advection equations and nonlinear Burgers' equations. In application problems, higher than third order WENO schemes are often preferred in order to efficiently resolve the complex solution structures. In this paper, we extend the approach to higher order WENO simulations specifically the fifth order WENO scheme. A fifth order WENO interpolation is applied in the prolongation part of the sparse-grid combination technique to deal with discontinuous solutions. Benchmark problems are first solved to show that significant CPU times are saved while both fifth order accuracy and stability of the WENO scheme are preserved for simulations on sparse grids. The fifth order sparse grid WENO method is then applied to kinetic problems modeled by high dimensional Vlasov based PDEs to further demonstrate large savings of computational costs by comparing with simulations on regular single grids.
