Linearly implicit predictor–corrector methods for space-fractional reaction–diffusion equations with non-smooth initial data

In this paper, linearly implicit predictor–corrector methods are proposed for solving space-fractional reaction–diffusion equations with non-smooth initial data. The methods are based on Matrix Transfer Technique for spatial discretization and are shown to be unconditionally stable. It is observed that the linearly implicit predictor–corrector method derived by using (1,1)-Padé approximation to matrix exponential function incurs oscillatory behavior for some time steps. These oscillations are due to high frequency components present in the solution and are diminished as the order of the space-fractional derivative decreases (slow diffusion). We present a priori reliability constraint to avoid these unwanted oscillations and generalize the constraints for all (m,m)-Padé approximants, m∈Z⁺, to the matrix exponential functions. These time stepping constraints are seen to be dependent on the order of the space-fractional derivative. The linearly implicit predictor–corrector method based on the (0,2)-Padé approximations to the matrix exponential function is shown to be oscillation-free for any time step. Error estimates are obtained for the methods and are theoretically shown to be second-order convergent. Computational complexity of the algorithms is discussed for solving multidimensional space-fractional reaction–diffusion systems. Several numerical experiments are performed to support our theoretical observations and to show the effectiveness, reliability, and efficiency of the methods.