Fast implicit integration factor method for nonlinear space riesz fractional reaction–diffusion equations

Abstract After spatially discretizing the nonlinear space Riesz fractional reaction–diffusion equations by the fractional centered difference formula, the resulting semi-discrete equations lead to a nonlinear ordinary differential equation system. In order to obtain good stability and robustness, the implicit integration factor (IIF) method that treats the reaction term implicitly and the diffusion term exactly is employed to solve the system. To deal with the extremely expensive cost in IIF, we also propose the shift-invert Lanczos method based on the Gohberg-Semencul formula to compute the matrix exponential, by using the particular property that the coefficient matrix is symmetric positive definite Toeplitz. The proposed fast method requires only O ( M ) memory storage and O ( M log M ) computation cost in each iterative step, compared to the O ( M 2 ) storage and O ( M 3 ) computational complexity of the direct solution method, where M is the size of the spatial grid. Some numerical experiments are performed to verify the correctness of the theoretical results and demonstrate the efficiency of our fast solution algorithm.