Numerical solution of advection–diffusion type equation by modified error correction scheme

Abstract

In this paper, we consider a numerical solution for nonlinear advection–diffusion equation by a backward semi-Lagrangian method. The numerical method is based on the second-order backward differentiation formula for the material derivative and the fourth-order finite difference formula for the diffusion term along the characteristic curve. A modified error correction scheme is newly introduced to efficiently find the departure point of the characteristic curve. Through several numerical simulations, we demonstrate that the proposed method has second and third convergence orders in time and space, respectively, and is efficient and accurate compared to existing techniques. In addition, it is numerically shown that the proposed method has good properties in terms of energy and mass conservation.