Numerical Analysis for the Model Governing Anomalous Dynamics in Expanding Media

Fourier stability analysis works well and is popular for the finite difference schemes of the linear partial differential equations. In some literature, the similar idea is also used to do the convergence analysis, but they require a strong assumption on the truncation error function, which is generally impossible to achieve. Here we provide an idea of strict Fourier convergence analysis and the assumption on the truncation error function is removed. Then we apply the idea of Fourier stability and convergence analyses to the finite difference schemes of two dimensional time fractional diffusion equation with time-dependent variable coefficients, in which the time fractional derivative is discretized by the L1 scheme on graded meshes to overcome the weak singularity of the solution at the initial time. Finally, the numerical experiments are performed to confirm the theoretical results.