Efficient Sparse-Grid Implementation of a Fifth-Order Multi-resolution WENO Scheme for Hyperbolic Equations

• Published in: Communications on Applied Mathematics and Computation (Online) Vol. 5; no. 4; pp. 1339 - 1364
• Main Authors: Tsybulnik, Ernie, Zhu, Xiaozhi, Zhang, Yong-Tao
• Format: Journal Article
• Language: English
• Published: Singapore Springer Nature Singapore 01.12.2023
• Subjects: Applications of Mathematics; Computational Science and Engineering; Mathematics; Mathematics and Statistics; Technical Note; Sparse grids; High spatial dimensions; 65M06; Multi-resolution WENO schemes; Hyperbolic partial differential equations (PDEs); Weighted essentially non-oscillatory (WENO) schemes; 35L65
• ISSN: 2096-6385; 2661-8893
• DOI: 10.1007/s42967-022-00202-4

High-order accurate weighted essentially non-oscillatory (WENO) schemes are a class of broadly applied numerical methods for solving hyperbolic partial differential equations (PDEs). Due to highly nonlinear property of the WENO algorithm, large amount of computational costs are required for solving multidimensional problems. In our previous work (Lu et al. in Pure Appl Math Q 14: 57–86, 2018; Zhu and Zhang in J Sci Comput 87: 44, 2021), sparse-grid techniques were applied to the classical finite difference WENO schemes in solving multidimensional hyperbolic equations, and it was shown that significant CPU times were saved, while both accuracy and stability of the classical WENO schemes were maintained for computations on sparse grids. In this technical note, we apply the approach to recently developed finite difference multi-resolution WENO scheme specifically the fifth-order scheme, which has very interesting properties such as its simplicity in linear weights’ construction over a classical WENO scheme. Numerical experiments on solving high dimensional hyperbolic equations including Vlasov based kinetic problems are performed to demonstrate that the sparse-grid computations achieve large savings of CPU times, and at the same time preserve comparable accuracy and resolution with those on corresponding regular single grids.