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.

加权的基本非振荡（WENO）方案，特别是五阶WENO方案，是求解双曲型偏微分方程（PDE）的一类流行的高阶精确数值方法。但是，当空间尺寸较大时，空间网格点的数量会大大增加。通过使用诸如五阶WENO方案的非线性高阶精度WENO方案，在数值模拟中导致大量的运算和计算成本。如何通过高阶WENO方法实现高空间维双曲线PDE的快速仿真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解决高维问题的一种非常有效的近似工具。在最近的著作中[陆，陈和张，《纯数学与应用数学季刊》，14（2018）57-86]，设计了一种采用稀疏网格组合技术的三阶有限差分WENO方法来求解多维双曲方程，包括线性对流方程和非线性Burgers方程。数值实验表明，在稀疏网格上的WENO计算与在常规单网格上的计算相比，在光滑的解区域和非线性稳定性方面具有可比的三阶精度。在应用程序问题中，通常优先选择高于三阶的WENO方案，以有效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仿真，特别是五阶WENO方案。在稀疏网格组合技术的扩展部分中应用了五阶WENO插值来处理不连续解。首先解决基准问题，以显示可节省大量CPU时间，同时保留了五阶精度和WENO方案的稳定性，可在稀疏网格上进行仿真。然后，将五阶稀疏网格WENO方法应用于基于高维Vlasov的PDE建模的动力学问题，以与常规的单个网格上的仿真相比，进一步证明可节省大量的计算成本。最后讨论了几个未解决的问题。然后，将五阶稀疏网格WENO方法应用于基于高维Vlasov的PDE建模的动力学问题，以与常规的单个网格上的仿真相比，进一步证明可节省大量的计算成本。最后讨论了几个未解决的问题。然后，将五阶稀疏网格WENO方法应用于基于高维Vlasov的PDE建模的动力学问题，以与常规的单个网格上的仿真相比，进一步证明可节省大量的计算成本。最后讨论了几个未解决的问题。