Efficient high-order compact exponential time differencing method for space-fractional reaction-diffusion systems with nonhomogeneous boundary conditions

This paper introduces an efficient unconditionally stable fourth-order method for solving nonlinear space-fractional reaction-diffusion systems with nonhomogeneous Dirichlet boundary conditions on bounded domains. The proposed method is based on a compact improved matrix transformed technique for fourth-order spatial approximation and exponential time differencing approximation for fourth-order time integration. The main advantage of the improved matrix transfer technique is that it leads to a system of ordinary differential equations with spatial discretization matrix raised to the desired fractional order. The key benefit of the fourth-order exponential integrator is that it can be implemented with essentially the same computational complexity as the backward Euler method by utilizing a partial fraction splitting technique in which it is just required to solve two backward Euler-type well-conditioned linear systems at each time step by computing LU decomposition of spatial discretization matrix once outside the time loop. Linear stability analysis and various numerical experiments are also performed to demonstrate stability and accuracy of the proposed method. Moreover, calculation of local truncation error and an empirical convergence analysis show the fourth-order accuracy of the proposed method.