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A rational radial basis function method for accurately resolving discontinuities and steep gradients. (English) Zbl 1397.65209

Summary: Radial Basis Function (RBF) methods have become important tools for scattered data interpolation and for solving partial differential equations (PDEs) in complexly shaped domains. When the underlying function is sufficiently smooth, RBF methods can produce exceptional accuracy. However, like other high order numerical methods, if the underlying function has steep gradients or discontinuities the RBF method may/will produce solutions with non-physical oscillations. In this work, a rational RBF method is used to approximate derivatives of functions with steep gradients and discontinuities and to solve PDEs with such solutions. The method is non-linear and is more computationally expensive than the standard RBF method. A modified partition of unity method is discussed as an way to implement the rational RBF method in higher dimensions.

MSC:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65D05 Numerical interpolation
41A05 Interpolation in approximation theory
41A20 Approximation by rational functions
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
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