In this paper, a conservative nonlinear implicit finite difference scheme for the generalized Rosenau-KdV equation is studied. Convergence and stability of the proposed scheme are proved by a discrete energy method. The proof with a priori error estimate shows that the convergence rates of numerical solutions are both the second order on time and in space. Meanwhile, numerical experiments are carried out to verify the theoretical analysis and show that the scheme is efficient and reliable.

We analyze a first order in time Fourier pseudospectral scheme for Swift-Hohenberg equation. One major challenge for the higher order diffusion non-linear systems is how to ensure the unconditional energy stability and we propose an efficient scheme for the equation based on the convex splitting of the energy. Theoretically, the energy stability of the scheme is proved. Moreover, following the derived aliasing error estimate, the convergence analysis in the discrete l 2-norm for the proposed scheme is given.