High accuracy compact difference and multigrid methods for two-dimensional time-dependent nonlinear advection-diffusion-reaction problems

We focus on high-accuracy and stable numerical methods for the time-dependent nonlinear advection-diffusion-reaction problems in this work. A novel fourth-order fully implicit compact difference scheme is derived, in which we make use of the fourth-order compact formulas to discretize the diffusion terms, the fourth-order Padé formulas to compute the nonlinear advection terms, and the fourth-order backward differencing formula to approximate the time derivative term. Convergence and stability of the new scheme are analysed by the energy method. On the basis of the novel scheme, a multigrid algorithm is established such that a time advancement algorithm for solving these nonlinear problems are obtained. We provide various numerical experiments to verify the performances of the proposed scheme including the reliability, stability and computational efficiency. And the results obtained by our method show it is more accurate than most same kind schemes reported in the literature.