Temporal second order difference schemes for the multi-dimensional variable-order time fractional sub-diffusion equations

A special point on each time interval is found for the approximation of the variable-order time Caputo derivative, which makes at least second order approximation accuracy be obtained. On this basis, two difference schemes are proposed for the multi-dimensional variable-order time fractional sub-diffusion equations, which have second order accuracy in time, second order and fourth order accuracy in space, respectively. The obtained difference schemes are proved to be uniquely solvable. The convergence and stability of the schemes in the discrete H₁-norm are analyzed by utilizing the energy method. Some numerical examples are presented to verify the theoretical results.