Error analysis of a fully discrete scheme for a multi-term time fractional diffusion equation with initial singularity

Abstract

In this paper, a fully discrete scheme for amulti-term time fractional diffusion equation with initialsingularity is considered. First, we construct a semi-discretescheme by using the $L1$ scheme on a graded mesh for the multi-termCaputo-type time fractional derivatives. It is shownthat with appropriate choice of the grading parameter $r$, themethod has the optimal $2-\\alpha_1$ order convergence, where$\\alpha_1~\\in~(0,1)$ is the highest fractional derivative order inthe multi-term time fractional derivatives. Then we design a fullydiscrete scheme by combining the spectral method for the spatialdiscretization. Convergence and unconditional stability are provedrigorously. In order to reduce the storage requirement andcomputational cost, a fast evaluation scheme for the time fractionalderivatives is applied. Numerical tests confirm that our erroranalysis is sharp and the fast algorithm improves the computationalefficiency significantly.