Superconvergent Non-Polynomial Approximations
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In this paper, we introduce a superconvergent approximation method using a non-polynomial that consists of radial basis functions (RBFs) to solve conservation laws. The use of RBFs for interpolation and approximation is a well developed area of research. Of particular interest in this work is the development of high order finite volume (FV) weighted essentially non-oscillatory (WENO) methods, which utilize RBF approximations to obtain required data at the cell interfaces. Superconvergence of the approximations is addressed through analyses, which result in the attainment of expressions for optimal RBF shape parameters. This study seeks to address the practical elements of the approach, including the evaluations of shape parameters and a hybrid implementation. To highlight the effectiveness of the non-polynomial basis, the proposed methods are applied to one-dimensional hyperbolic and weakly hyperbolic systems of conservation laws. In the latter case, notable improvements are observed in predicting the location and height of the finite time blowup. The convergence results demonstrate that the proposed schemes attain improvements in accuracy, as indicated by the analyses.
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