DMS Applied Mathematics Seminar

Speaker: Todd Arbogast, University of Texas at Austin

Title: Polynomial and RBF Finite Volume WENO Reconstructions for Solving Conservation Laws

Abstract: We analyze weighted essentially nonoscillatory (WENO) reconstructions of functions from finite volume average values, basing the approximations on either polynomials or radial basis functions (RBFs). In the polynomial case, we present new multi-level WENO reconstructions with adaptive order (WENO-AO). We give conditions under which the polynomial reconstructions achieve optimal order accuracy for both smooth solutions and solutions with discontinuities. The theory of existence of finite volume RBF approximations is developed, and numerical evidence is given to show that the RBF approximation converges 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. These WENO-AO and RBF-WENO-AO reconstructions are applied to develop finite volume schemes for hyperbolic conservation laws on nonuniform meshes over multiple space dimensions. Numerical examples show that the schemes maintain proper accuracy and achieve the essentially non-oscillatory property when solving hyperbolic conservation laws.

Joint work with Chieh-Sen Huang, Ming-Hsien Kuo, and Xikai Zhao