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Essentially non-oscillatory (ENO) and weighted ENO (WENO) methods are efficient high order numerical methods for solving hyperbolic conservation laws designed to reduce the Gibbs oscillations. The original ENO and WENO methods are based on the polynomial interpolation and the overall convergence rate is uniquely determined by the total number of interpolation points involved for the approximation. In this paper, we propose non-polynomial ENO and WENO finite volume methods in order to enhance the local accuracy and convergence. The infinitely smooth radial basis functions (RBFs) are adopted as a non-polynomial interpolation basis. Particularly we use the multi-quadratic and Gaussian RBFs. The non-polynomial interpolation such as the RBF interpolation offers the flexibility to control the local error by optimizing the free parameter. Then we show that the non-polynomial interpolation can be represented as a perturbation of the polynomial interpolation. To guarantee the essentially non-oscillatory property, the monotone polynomial interpolation method is introduced as a switching method to the polynomial reconstruction adaptively near the non-smooth area. The numerical results show that the developed non-polynomial ENO and WENO methods with the monotone polynomial interpolation method enhance the local accuracy and give sharper solution profile than the ENO/WENO methods based on the polynomial interpolation.
Journal: Applied Numerical Mathematics - Volume 112, February 2017, Pages 27–50