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