A compact fourth-order [formula omitted]-stable scheme for reaction–diffusion systems with nonsmooth data

Abstract

Exponential time differencing Runge–Kutta (ETDRK) schemes based on diagonal Padé approximations for the numerical solution of reaction–diffusion systems containing nonsmooth data have the disadvantage of producing poor numerical results when the time steps are “too large” relative to the spatial steps. On the other hand, methods based on sub-diagonal Padé approximations do not suffer from this bottleneck. In this paper, we present a novel modified version of Cox and Matthews fourth-order ETDRK scheme based on sub-diagonal Padé approximation to matrix exponential functions in combination with compact fourth-order finite difference scheme (in space) for direct integration of reaction–diffusion problems with nonsmooth data. For an efficient implementation of the scheme, a partial fraction splitting technique is utilized in which it is required to solve several backward Euler-type linear systems at each time step. Moreover, the design of the algorithm offers parallel implementation, so we implement the proposed algorithm in parallel on two processors utilizing MatlabMPI (a parallel, message passing version of Matlab) and obtain that the parallel version is computationally more efficient than the existing schemes considered in this paper. We investigate the amplification factor of the scheme and plot its boundaries of stability regions which give an indication of the stability of the scheme. Calculation of the local truncation error and an empirical convergence analysis demonstrates the fourth-order accuracy of the proposed scheme. Accuracy, computational efficiency, and reliability of the new scheme are demonstrated with numerical examples and comparing it with existing schemes.