New local radial point interpolation-FD methods for solving fractional diffusion and damped-wave problems

Abstract

A new approach is proposed for developing numerical schemes for the solution of fractional diffusion and diffusion-wave equations. The novelty lies in the construction of analytical shape functions in a three-point local radial point interpolation method. The gain is that numerical computations of matrix entries are not required and as in the finite difference setting, the availability of matrix coefficients in closed-form facilitates stability and convergence analyses. For the fractional diffusion equation, a weak-form formulation is used to develop an implicit meshless finite difference method. The unconditional stability and convergence of the scheme is theoretically established and numerical examples are given to illustrate its second-order accuracy in space. The expected time convergence and the dependence of the convergence on the multiquadric shape parameter are also investigated. The technique is applied to develop numerical schemes for the fractional diffusion-wave equation. For this problem, the new scheme is shown to perform well in terms of both temporal and spatial accuracies when compared to some existing methods.