Multiquadric based RBF-HFD approximation formulas and convergence properties

Abstract

RBF-FD formulas have received special attention due to their superior accuracy, well-conditioning, and ease of implementation in solving partial differential equations. To further improve the accuracy of RBF-FD formulas, without increasing the stencil size RBF-HFD formulas are proposed in the literature. In this article, we obtain leading/higher order analytical expressions for weights and local truncation errors associated with various such formulas using the multiquadric radial function. We also study the convergence properties of these formulas. Symbolic computation in Mathematica is employed for analytically solving linear systems for the unknown weights in the RBF-HFD formulas. Symmetry properties of central difference approximations help reduce the number of unknown weights and thereby the size of the linear system. We compute the analytical expressions of weights and local truncation errors for first and second-derivative approximation formulas till tenth order; and for two-dimensional Laplacian operator approximation formulas till sixth order. The obtained formulas are free from ill-conditioning. It is observed that the RBF-HFD formula weights converge to corresponding order classical compact FD scheme weights when the shape parameter (ϵ) tends to zero. The local truncation error has global minimum for an optimal value of the shape parameter. Further, the optimal shape parameter is independent of nodal distance but depends on choice of test function and its derivative values at a reference node.