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

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. 2018, Zhu and Zhang 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 thesis, 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.