Numerical Methods for Solution of Partial Differential Equations

Numerical Methods for Solution of Partial Differential Equations 

Simulation methods for spatially extended systems are much more numerically challenging than are those for homogeneous systems. The governing equations, that is, the reaction-diffusion equations, are systems cff partial dif- 

We consider a general two-variable model involving reaction and diffusion, The PDE system is given by the following pair of equations  

A similar expression holds for the values of v at each grid point, that is, for v j. We need to discretize the derivatives in time, as well, and this can be done in either an implicit or an explicit fashion. Considering first, for simplicity, an explicit Euler algorithm for the discretized time derivatives, the following system of equations will result  

This is sometimes done to enhance numerical stability because implicit schemes provide a source of feedback, as was described previously in the section on numerical methods for ODEs. If this approach is taken, the resulting equations must be solved for the (n + l)th values of the variables. An example will illustrate how this is done (the following presentation is taken from Ref. 51). 

---