Time-Step Constraints for Reaction-Diffusion Problems
Fortunato, Daniel F.
- Reaction-diffusion equations naturally arise in many biological, physical, chemical, and ecological contexts. Fisher's equation is the simplest reaction-diffusion equation. It models a population growing logistically while diffusing in space and admits traveling wave solutions. We consider finite difference schemes for Fisher's equation, which depend on the size of the time step used. We can ... read morediscretize Fisher's equation using explicit methods (like forward Euler or midpoint) or implicit methods (like backward Euler or Crank-Nicolson). For the explicit methods, a time step constraint appears as the result of a CFL condition. For backward Euler, there is no time step constraint. However, to solve the nonlinear system of equations with Newton's method or fixed-point iteration, the constraint reappears as a convergence constraint. If instead we use multigrid to solve the nonlinear system of equations, the constraint comes back as a convergence constraint for both the correction scheme (CS) and the full approximation scheme (FAS). If in the relaxation scheme for FAS we solve the nonlinear equations exactly---which the simplicity of Fisher's equation allows us to do---then the constraint appears once again as an accuracy constraint. Crank-Nicolson is unconditionally stable and second-order in time, so discretizing Fisher's equation using Crank-Nicolson and then using FAS to solve the system of equations relaxes the accuracy constraint somewhat.read less