Numerical integration methods for a nonlinear diffusion equation

Abstract EN

To solve a diffusion-reaction type equation one can use several finite difference schemes.

In general the standard explicit methods are known to be unstable and a high price is paid for the implicit method due to the inversion of the large matrices involved.

Furthermore, the method is prohibitive in more than two dimensions due to restrictions on memory and large computation times.

On the other hand, if the partial differential equation is not linear, then one needs to solve large systems of nonlinear algebraic equations which is a troublesome business.

In this paper an explicit method is presented and shown to be stable for both decreasing in time solutions and blow-up solutions of the given initial boundary value problem.

An implicit Crank-Nicolson scheme is also presented and shown to be stable for both kind of solutions.

However, more accurate results are expected from this latter scheme since the O(Ar) terms in the growth factor are bounded independently of the solution.

The initial-boundary value problem is solved by using the two schemes.

The results are in agreement and show the existence of the blow-up solutions of the problem.