Fourth-order time stepping methods with matrix transfer technique for space-fractional reaction-diffusion equations

Abstract

In this paper, we develop two high-order methods for space-fractional reaction-diffusion equations. The schemes are based on fourth-order Matrix Transfer Technique (MTT) for spatial discretization and fourth-order Exponential Time Differencing Runge–Kutta (ETDRK) for temporal discretization. The ETDRK schemes are based on diagonal and sub-diagonal Padé approximations to the matrix exponential functions. It is observed that the A-stable scheme incurs unwanted oscillations due to high-frequency components present in the solution. These oscillations diminish as the order of the space-fractional derivative decreases. We propose a reliability constraint which is dependent on the order of the space-fractional derivative to avoid these oscillations. However, the L-stable scheme is oscillation-free for any time step. Partial fraction splitting technique is used to compose computationally efficient versions of the schemes. The amplification factor of the schemes is investigated by plotting their stability regions. Convergence analysis is performed on numerical experiments to demonstrate the fourth-order accuracy of the developed schemes in space and time. Numerical experiments made on multidimensional space-fractional reaction-diffusion equations show the reliability, stability, and efficiency of the schemes.