Conservative compact difference scheme based on the scalar auxiliary variable method for the generalized Kawahara equation

In this paper, a conservative compact difference scheme for the generalized Kawahara equation is constructed based on the scalar auxiliary variable (SAV) approach. The discrete conservative laws of mass and Hamiltonian energy and boundedness estimates are studied in detail. The error estimates in discrete $L^{\infty}$-norm and $L^2$-norm of the presented scheme are analyzed by using the discrete energy method. We give an efficiently algorithm of the presented scheme which only needs to solve two decoupled equations.

A fast compact difference scheme for the fourth-order multi-term fractional sub-diffusion equation with non-smooth solution

In this paper, we develop a fast compact difference scheme for the fourth-order multi-term fractional sub-diffusion equation with Neumann boundary conditions. Combining L1 formula on graded meshes and the efficient sum-of-exponentials approximation to the kernels, the proposed scheme recovers the losing temporal convergence accuracy and spares the computational costs. Meanwhile, difficulty caused by the Neumann boundary conditions and fourth-order derivative is also carefully handled. The unique solvability, unconditional stability and convergence of the proposed scheme are analyzed by the energy method. At last, the theoretical results are verified by numerical experiments.