Adaptive WENO methods based on radial basis functions reconstruction
We explore the use of radial basis functions (RBF) in the weighted essentially non-oscillatory (WENO) reconstruction process used to solve hyper- bolic conservation laws, resulting in a numerical method of arbitrarily high order to solve problems with discontinuous solutions. Thanks to the mesh-less property of the RBFs, the method is suitable for non uniform grids and mesh adaptation. We focus on multiquadric radial basis functions and propose a simple strategy to choose the inherent shape parameter to control the balance between theoret- ical achievable accuracy and the numerical stability. We also develop an original smoothness indicator independent of the chosen RBF for the WENO reconstruc- tion step. Moreover, we introduce type I and type II RBF-WENO methods by computing specific linear weights. The RBF-WENO method is used to solve linear and nonlinear problems for both scalar conservation laws and systems of conserva- tion laws, including Burgers equation, the Buckley-Leverett equation, and the Eu- ler equations. Numerical results confirm the performance of the proposed method. We finally consider an effective conservative adaptive algorithm that captures mov- ing shocks and rapidly varying solutions well. Numerical results on moving grids are presented for both Burgers equation and the more complex Euler equations.