The Time Second Order Mass Conservative Characteristic FDM for Advection–Diffusion Equations in High Dimensions

Abstract

We propose in this paper a time second order mass conservative algorithm for solving advection–diffusion equations. A conservative interpolation and a continuous discrete flux are coupled to the characteristic finite difference method, which enables using large time step size in computation. The advection–diffusion equations are first transformed to the characteristic form, for which the integration over the irregular tracking cells at previous time level is proposed to be computed using conservative interpolation. In order to get second order in time solution, we treat the diffusion terms by taking the average along the characteristics and use high order accurate discrete flux that are continuous at tracking cell boundaries to obtain mass conservative solution. We demonstrate the second order temporal and spatial accuracy, as well as mass conservation property by comparing results with exact solutions. Comparisons with standard characteristic finite difference methods show the excellent performance of our method that it can get much more stable and accurate solutions and avoid non-physical numerical oscillation.