Pointwise-in-time error estimate of an ADI scheme for two-dimensional multi-term subdiffusion equation

Abstract

We use alternating direction implicit (ADI) difference method to solve a two-dimensional multi-term time-fractional subdiffusion equation with Dirichlet boundary conditions whose solution has a typical weak singularity at the initial time. L1 scheme on uniform mesh is used to discretize the multi-term temporal Caputo fractional derivatives with orders $\alpha _l \in (0,1)$ for $l=1,2,\dots J$. The sharp pointwise-in-time error estimate is given for the fully discrete ADI scheme, and the final error bound is $\alpha _1$-robust (i.e. the error bound does not blow up when $\alpha _1\rightarrow 1^-$), where $\alpha _1$ is the highest fractional derivative order in the multi-term time fractional derivatives. Through theoretical analysis and numerical experiments, we prove that the convergence order of the fully discrete ADI scheme is $O(\tau ^{2\alpha _{1}}+\tau t_{n}^{\alpha _{1}-1}+h_1^2+h_2^2)$ at time $t=t_n$; therefore, the temporal accuracy of the scheme can attain $O(\tau ^{\min \{2\alpha _{1},1\}})$ when t is away from 0, while globally it is $O(\tau ^{\alpha _{1}})$.