Solving a non-linear fractional convection-diffusion equation using local discontinuous Galerkin method

Abstract

We propose a local discontinuous Galerkin method for solving a nonlinear convection-diffusion equation consisting of a fractional diffusion described by a fractional Laplacian operator of order 0<p<2, a nonlinear diffusion, and a nonlinear convection term. The algorithm is developed by the local discontinuous Galerkin method using Spline interpolations to achieve higher accuracy. In this method, we convert the main problem to a first-order system and approximate the outcome by the Galerkin method. In this study, in contrast to the direct Galerkin method using Legender polynomials, we demonstrate that the proposed method can be suitable for the general fractional convection-diffusion problem, remarkably improve stability and provide convergence order O(hk+1), when k indicates the degree of polynomials. Numerical results have illustrated the accuracy of this scheme and compare it for different conditions.