Order enhanced finite volume methods through non-polynomial approximation

In this paper, we introduce an approximation method that establishes certain order enhancements by leveraging radial basis functions (RBFs) in the numerical solution of 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 cell interfaces. The aforementioned improvement in the order of accuracy is addressed through an analysis of the truncation error, resulting in expressions for the shape parameters appearing in the basis. This paper seeks to address the practical elements of the approach, including the evaluations of shape parameters as well as a hybrid implementation. To highlight the effectiveness of the non-polynomial basis in shock-capturing, the proposed methods are applied to systems of one-dimensional hyperbolic and weakly hyperbolic conservation laws and compared with several well-known WENO schemes in the literature. We also include a two-dimensional example for a scalar problem that demonstrates an extension to multiple dimensions. In the case of the non-smooth, weakly hyperbolic test problem, notable improvements are observed in predicting the location and height of the finite time blowup. The numerical results demonstrate that the proposed schemes attain notable improvements in accuracy, as indicated by the analysis of the reconstructions. A key contribution of this work is the development of robust third-order WENO method, which further demonstrates the effectiveness of the non-polynomial basis.