A fast and high-order IMEX method for non-linear time-space-fractional reaction-diffusion equations

An efficient and high-order numerical method is presented for solving a time-space-fractional reaction-diffusion equation. Matrix transfer technique based on fourth-order compact finite differences is first used to discretize the space-fractional Laplacian operator which results in a system with a linear stiff term. Then, an implicit-explicit (IMEX) trapezoidal product-integration rule is implemented for time integration which treats the stiff linear term implicitly and non-linear non-stiff term explicitly. The stability and convergence of the method are analyzed. Due to the discontinuity of the solution derivative at $$t=0$$ , the numerical method is only $$1+\alpha $$ order accurate in time where $$\alpha $$ is the order of the time-fractional derivative. Richardson extrapolation is introduced to obtain a modified version of the method which is second order accurate in time. A fast algorithm based on discrete sine transform is also implemented to reduce the cost of computing the discretized space-fractional Laplacian operator.