Fast Sparse Grid Simulations of Fifth Order WENO Scheme for High Dimensional Hyperbolic PDEs

Zhu Xiaozhi; Zhang Yong-Tao

摘要

The weighted essentially non-oscillatory (WENO) schemes, especially the fifth order 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. Numerical experiments showed that WENO computations on sparse grids achieved comparable third order accuracy in smooth regions of the solutions and nonlinear stability as that for computations on regular single grids. 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. Several open problems are discussed at last.

机译：基本上非振荡（WENO）方案，特别是第五阶WENO方案的加权是一种流行的高阶准确数值方法，用于求解双曲线部分微分方程（PDE）。然而，当空间尺寸高时，空间网格点的数量显着增加。通过使用非线性高阶精度Weno方案（例如第五阶Weno方案），它在数值模拟中导致数值模拟中的大量操作和计算成本。如何通过高阶Weno方法实现快速模拟，用于高空间尺寸双曲线PDE是一个具有挑战性和重要的问题。在文献中，稀疏电网技术已被开发为高尺寸问题的非常有效的近似工具。在最近的工作[卢，陈和张，纯净和应用数学季度，14（2018）57-86]中，具有稀疏 - 网格组合技术的三阶有限差异Weno方法，旨在解决包括线性平流的多维双曲方程方程和非线性汉堡方程。数值实验表明，Weno计算在稀疏区域中实现了可比的第三顺序精度，在解决方案的平滑区域和非线性稳定性，以及用于普通单网格上的计算。在应用问题中，通常优选高于三阶Weno方案，以便有效地解决复杂的解决方案结构。在本文中，我们特别延长了更高阶Weno模拟的方法，特别是第五阶Weno方案。在稀疏 - 网格组合技术的延长部分中应用第五阶文核插值，以处理不连续解决方案。首先解决了基准问题，以显示保存了显着的CPU时间，而Weno方案的第五顺序精度和稳定性保存用于稀疏网格上的模拟。然后将第五阶稀疏电网Weno方法应用于由高维Vlasov基于高维Vlasov的PDE建模的动力学问题，以进一步展示通过与常规单网格上的模拟进行比较来节省大量计算成本。最后讨论了几个公开问题。

Fast Sparse Grid Simulations of Fifth Order WENO Scheme for High Dimensional Hyperbolic PDEs

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