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

Abstract

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 kernel, 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.