High-order Compact Difference Schemes For Solving Unsteady Convection Diffusion Reaction Equations

In this paper,a fourth-order compact finite difference scheme for solving the unsteady convection diffusion reaction equations is established.Firstly,a fourth-order compact difference equation is used for the spatial derivative term,the higher-order derivative term involved in the derivation process is discretized by the method of derivation of the original equation,and the fourth-order backward Euler formula is used for the time derivative term.A five-layer unconditionally stable and high-order compact scheme for solving the one-dimensional(1D)unsteady convection diffusion reaction equation is obtained.This scheme has fourth-order accuracy both in time and space,then several numerical examples with exact solutions are used to verify the theoritical results.Compared with the numerical results of the existing numerical methods in the reference,the superiority and stability of the proposed scheme are validated.Then,the ID high-order compact difference method is directly extended to the 2D and 3D problems.For 2D and 3D,iterative calculations are required.Therefore,a modified multigrid full approximation scheme is used to speed up the iteration speed,reduce the number of iterations,save computing time,and improve computational efficiency.Finally,some numerical examples with exact solutions are used to verify the theoritical results.The numerical results show that the present method in this paper can fully achieve the fourth-order accuracy in both time and space.And the computed error is obviously smaller than the numerical results in the literature.This is also the biggest advantage of the scheme of this paper compared to other schemes in the literature.The equation studied in this paper is general,especially when dealing with large Reynolds number,it still has certain advantages,and when calculating nonlinear problems,the computed results of the present scheme are still more accurate than those in the literature.Therefore,the accuracy,stability and high efficiency of this paper in the numerical solution of unsteady convection diffusion reaction equations are verified.