Weighted essentially non-oscillatory generalized finite difference method for hyperbolic conservation laws

Abstract

This paper introduces a weighted essentially non-oscillatory (WENO) generalized finite difference method (GFDM) for solving hyperbolic conservation laws. The proposed method accommodates both unstructured meshes and scattered point clouds, ensuring the preservation of free-stream solutions. Drawing from an alternative WENO scheme formulation, the flux functions are disassembled into low-order and high-order constituents. Diverging from conventional finite difference methods, the WENO reconstruction is executed directly on the numerical solutions, rather than the flux functions, with Riemann solvers employed exclusively for the low-order components, thereby diminishing computational overhead. This paper outlines the process for stencil selection, WENO reconstruction, and scheme formulation. Numerical examples for one- and two-dimensional conservation laws validate the high-order accuracy and non-oscillatory properties of the method.