Comparison of the nodal integral method and nonstandard finite-difference schemes for the Fisher equation

The relationship between the so-called nonstandard finite-difference schemes and the nodal integral method (NIM) is investigated using the Fisher equation as a model problem. Exact and best finite-difference schemes are reviewed first. Next, the NIM for the Fisher equation is developed. It is shown that the NIM leads to a nonstandard evaluation of the derivatives. Moreover, the resulting scheme possesses the desirable characteristics of the nonstandard finite difference schemes, such as the nonlocal evaluation of the nonlinear terms. Thus, the NIM provides a systematic framework to obtain schemes similar to the best finite-difference schemes. Numerical results for a propagating front problem show that the NIM can capture the shape and speed of the front very accurately. Results also show that the best finite-difference scheme is stable for large grid sizes but only at the cost of inaccuracy in the front propagation speed. Additional results are obtained using the NIM for symmetric and asymmetric initial conditions. These describe the interaction of two fronts of advantageous genes that approach each other and merge. It is noted that the traveling fronts evolving from certain asymmetric initial conditions become indistinguishable from those that evolve from symmetric initial conditions.