Adaptive Radial Basis Function ENO and WENO Methods for Hyperbolic Conservation Law

A hyperbolic conservations law is a hyperbolic partial differential equation which describes the movement of conserved quantities. A special property of hyperbolic conservation laws is that discontinuities could be developed even though a smooth initial condition is used. If that is the case, high order numerical methods would suffer from the Gibbs phenomenon that spurious oscillations appear near the discontinuities. In 1987, the Essentially Non-Oscillatory (ENO) scheme was proposed as the first uniformly high order numerical scheme that gives non-oscillatory approximation even around the discontinuities. From 1994 to 1996, the Weighted Essentially Non-Oscillatory (WENO) was constructed based on the ENO scheme that achieves higher order of accuracy within smooth region. Since then, variations of ENO/WENO schemes have been developed. But as the classical ENO/WENO schemes, most variations are still based on the polynomial interpolation.;The polynomial interpolation is widely used in many scientific disciplines as a way to interpolate scattered data points. But interpolating with non-polynomial basis has always been an active research area. In 1968, the Multiquadric method, which is a special case of Radial Basis Function (RBF) method was developed. Since then, different types of RBF basis have been proposed. Compared to the polynomial basis, the RBF basis at any point depends only on the distance between that point and the grid points. Therefore, the RBF interpolation can efficiently handle unstructured data points. In 2010, this property has been used to construct the ADER method which extends the ENO/WENO scheme to problems with non-uniform resolution.;In this thesis we are more interested in the other property that some RBFs contain unknown shape parameters. In other words, we want to adopt the RBF interpolation for the ENO/WENO reconstruction and optimize the parameters so that the new RBF-ENO/WENO method performs better in terms of accuracy and convergence. The RBF-ENO/WENO method is constructed in a way such that only a small modification of the existing ENO/WENO algorithm is required, which makes it convenient to switch between those two methods. Replacing the polynomial interpolation with the RBF interpolation is not consistent. In order to recover at least ENO/WENO performance in non-smooth region, a new edge detection method called monotone method is developed. Compared to existing edge detection methods, this method is more efficient in recognizing the non-smooth region locally. Numerical experiments for one and two dimensional problems are included to show that the RBF-ENO/WENO method outperforms the classical ENO/WENO schemes in both smooth and non-smooth region. Future directions related to this project are also included.