Consistent, non-oscillatory RBF finite difference solutions to boundary layer problems for any degree on uniform grids

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. (C) 2020 Elsevier Ltd. All rights reserved.

Automated summary

This paper introduces the use of Gaussian radial basis function finite difference method to solve singular perturbation problems on uniform grids, demonstrating the possibility of achieving non-oscillatory and accurate solutions through the adaptivity of the shape parameter, even with a small number of grid points. The optimal value of the shape parameter can lead to non-oscillatory and accurate solutions, and in some cases, the error can vanish completely. The paper also suggests choosing the optimal shape parameter to minimize error using the total variation norm of the numerical solution.