Finite Difference Schemes Satisfying Two Conservation Laws For Linear Advection Equation And KdV Equation

In recent years, Mao and his co-workers developed difference schemes for linear advection equation which satisfied two or three discrete conservation laws, see [44], [45], [15], [48] and [49]. The numerical results of the developed schemes were far better than traditional difference schemes' at both solutions' accuracy and long-time numerical integrations.The first work in this paper is to do the numerical analysis for the difference scheme staisfing two discrete conservation laws for the linear advection equation. We reveal that numerical errors of the scheme at successive time steps cancel with each other, which is a feature that is rarely seen in the numerical methods we have ever known. It is this feature of error-self-canceling that makes our scheme far better than traditional schemes.The second work in this paper is to apply the numerical approach developed for the linear advection equation to the KdV equation. We develop a scheme for the KdV equation, which satisfies the first two conservation laws of the equation. In constructing the scheme, we adopt the splitting strategy to split the equation into the conservation part and the dispersion part, and the approach of designing scheme satisfying two conservation laws is applied in the discretization of the conservation part. The developed scheme shows good quality in long-time numerical simulations.