On the numerical solution of Fisher’s equation with coefficient of diffusion term much smaller than coefficient of reaction term

Abstract

Li et al. (SIAM J. Sci. Comput. 20:719–738, 1998) used the moving mesh partial differential equation (MMPDE) to solve a scaled Fisher’s equation and the initial condition consisting of an exponential function. The results obtained are not accurate because MMPDE is based on a familiar arc-length or curvature monitor function. Qiu and Sloan (J. Comput. Phys. 146:726–746, 1998) constructed a suitable monitor function called modified monitor function and used it with the moving mesh differential algebraic equation (MMDAE) method to solve the same problem of scaled Fisher’s equation and obtained better results. In this work, we use the forward in time central space (FTCS) scheme and the nonstandard finite difference (NSFD) scheme, and we find that the temporal step size must be very small to obtain accurate results. This causes the computational time to be long if the domain is large. We use two techniques to modify these two schemes either by introducing artificial viscosity or using the approach of Ruxun et al. (Int. J. Numer. Methods Fluids 31:523–533, 1999). These techniques are efficient and give accurate results with a larger temporal step size. We prove that these four methods are consistent for partial differential equations, and we also obtain the region of stability.