Adaptive WENO Methods Based on Radial Basis Function Reconstruction

Adaptive WENO Methods Based on Radial Basis Function Reconstruction We explore the use of radial basis functions (RBF) in the weighted essentially non-oscillatory (WENO) reconstruction process used to solve hyperbolic 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 shape parameter to control the balance between achievable accuracy and the numerical stability. We also develop an original smoothness indicator which is independent of the RBF for the WENO reconstruction 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 and systems of conservation laws, including Burgers equation, the Buckley–Leverett equation, and the Euler equations. Numerical results confirm the performance of the proposed method. We finally consider an effective conservative adaptive algorithm that captures moving shocks and rapidly varying solutions well. Numerical results on moving grids are presented for both Burgers equation and the more complex Euler equations. Journal of Scientific Computing Springer Journals

Adaptive WENO Methods Based on Radial Basis Function Reconstruction

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Springer US
Copyright © 2017 by Springer Science+Business Media New York
Mathematics; Algorithms; Computational Mathematics and Numerical Analysis; Mathematical and Computational Engineering; Theoretical, Mathematical and Computational Physics
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