A Compact Difference Scheme for Time-Fractional Dirichlet Biharmonic Equation on Temporal Graded Meshes

Abstract

The stability of a compact finite difference scheme on general nonuniform temporal meshes for a time fractional two-dimensional biharmonic problem is proved and graded mesh error estimates are derived. By using the Stephenson scheme for spatial derivatives discretisation, we simultaneously obtain approximate values of the gradient without any loss of accuracy. The discretisation of the Caputo derivative on graded meshes leads to a fully discrete implicit scheme. Numerical experiments support the theoretical findings and indicate that for problems with nonsmooth solutions, graded meshes have an advantage for very coarse temporal meshes.