RBF WENO Reconstructions with Adaptive Order and Applications to Conservation Laws

摘要

Abstract This paper develops a framework for finite volume radial basis function (RBF) approximation of a function u on a stencil of mesh cells in multiple dimensions. The theory of existence of the approximation is given. In one dimension, as the cell diameters tend to zero, numerical evidence is given to show that the RBF approximation converges to u to the same order as a polynomial approximation when the RBF is infinitely differentiable. Specific multiquadric RBFs on stencils of 2 and 3 mesh cells are proven to have this convergence property. A two-level RBF based weighted essentially non-oscillatory (WENO) reconstruction with adaptive order (RBF-WENO-AO) is developed. WENO-AO reconstructions use arbitrary linear weights, and so they can be developed easily for RBF approximations, even on nonuniform meshes in multiple dimensions. Following the classical polynomial based WENO, a smoothness indicator is defined for the reconstruction. For one dimension, the convergence theory is given regarding the cases when u is smooth and when u has a discontinuity. These reconstructions are applied to develop finite volume schemes for hyperbolic conservation laws on nonuniform meshes over multiple space dimensions. The focus is on reconstructions based on multiquadric RBFs that are third order when the solution is smooth and second order otherwise, i.e., RBF-WENO-AO(3,2). Numerical examples show that the scheme maintains proper accuracy and achieves the essentially non-oscillatory property when solving hyperbolic conservation laws.