Numerical solution of a nonlinear reaction-diffusion equation

A nonlinear reaction-diffusion equation is studied numerically by a Petrov-Galerkin finite element method, which has been proved to be 2nd-order accurate in time and 4th-order in space. The comparison between the exact and numerical solutions of progressive waves shows that this numerical scheme is quite accurate, stable andefflcient. It is also shown that any local disturbance will spread, have a full growth and finally form two progressive waves propagating in both directions. The shape and the speed of the long term progressive waves are determined by the system itself, and do not depend on the details of the initial values.