Consistent, non-oscillatory RBF finite difference solutions to boundary layer problems for any degree on uniform grids. (May 2021)

• Record Type: Journal Article
• Authors: Gu, Jiaxi; Jung, Jae-Hun
• Is Part Of: Applied mathematics letters. Volume 115(2021)
• Publication Date: 2021-05
• Subjects: Singular perturbation problem -- Finite difference method -- Gaussian radial basis function -- Shape parameter; Applied mathematics -- Periodicals; 519.05
• DOI: 10.1016/j.aml.2020.106944
• Languages: English
• ISSNs: 0893-9659

Abstract

Abstract: Singular perturbation problem cannot be solved without oscillations by the classical finite difference methods on the uniform grid unless sufficient resolution is employed. In this note we use the Gaussian radial basis function (RBF) finite difference method and show that it is possible for the finite difference method to yield non-oscillatory and accurate solutions to the problem for any degree of resolution on uniform grids with the adaptivity of the shape parameter of the RBFs. Even with a highly small number of grid points, non-oscillatory and accurate solutions to the singular perturbation problem can be obtained with the optimal value of the shape parameter. We show that even the error vanishes for some cases. We also provide a possible suggestion to choose the optimal shape parameter that minimizes the error using the total variation norm of the numerical solution.