Two boundedness and monotonicity preserving methods for a generalized Fisher-KPP equation

A semi-explicit finite difference method and an implicit finite difference method are proposed for a generalized Fisher-KPP equation with two space variables. It is proved that these two methods could preserve the skew-symmetry of this equation. Moreover, under a condition on the step sizes, the semi-explicit method is capable of preserving the positivity, the boundedness, and the spatial and temporal monotonicity of initial approximations. And the implicit method is able to preserve these properties with no restriction on the step sizes. The stability and convergence of these two methods are also analyzed respectively. Finally, some numerical simulations are provided to verify the validity of our analytical results.