An advanced numerical modeling for Riesz space fractional advection–dispersion equations by a meshfree approach

Abstract

The space fractional advection–dispersion equations (SFADE) have been found to be very adequate in describing anomalous transport and dispersion phenomena. Aside from enjoying huge advantage in modeling, we have to face severe challenge presented by the non-locality of space fractional order derivatives, which is difficult to be handled by the traditional finite difference method (FDM) especially on a complex domain with irregularly distributed nodes. Therefore, it is crucial to develop a powerful numerical method to overcome this barrier.In this paper, the point interpolation method (PIM), a meshfree method, is further developed to solve SFADE, where the polynomial point-interpolation functions and their fractional derivatives with explicit expressions are substituted into Galerkin weak form of SFADE to obtain the discrete approximation system. Adopting an expanding technique, we develop an extended point interpolation method for SFADE with zero Dirichlet boundary condition. This technique avoids singular integral and turns system matrix into Toeplitz matrix. An efficient iteration algorithm is also suggested to reduce computing time and storage. Numerical experiments is presented to validate the newly developed method and to investigate accuracy and efficiency.