Fourth order RBF-HFD method for elliptic PDE problems with optimal shape parameter selection : Sixth order convergence and improved accuracy

Abstract

We develop fourth order radial basis function (RBF) based compact FD type methods for solving boundary value problems on elliptic PDEs in multi-dimensions. Traditional compact FD methods are derived using polynomial basis functions in Hermite interpolation based finite difference (HFD) method. The proposed method depends on inverse multiquadric (IMQ) RBFs with a free shape parameter. Analytical formulas for approximating first and second derivatives along with expressions for their local truncation errors are obtained in terms of the shape parameter and inter-nodal distance. We discuss the consistency, stability and convergence aspects of the discretization schemes for the boundary value problems. We also discuss optimal shape parameter selection strategies based on optimization of cost functions defined in terms of maximum absolute error, local truncation error and global truncation error. Using standard example problems on Poisson and Helmholtz equations, we demonstrate sixth order convergence and improved accuracy in the numerical solutions obtained by the IMQ-HFD method with shape parameter optimization. In two, three and four dimensions, we make use of tensor products and its properties for fast implementation of the discretization schemes. For solving multi-dimensional problems on large grid sizes, we harness the power of GPU computing to significantly accelerate the computation.