Hairer & Wanner’s Brusselator problem

A good test of the Rosenbrock methods can be applied to the PDE from Hairer & Wanner. The PDE’s for the diffusion equation (also known as the Brusselator in one spatial coordinate) are:

 

The initial & boundary conditions are:

The first step in the solution of this PDE is to convert it into equivalent ODE’s, with the use of the finite difference technique & the MOL. Here is the procedure to solve the PDE’s:

• Choose the number of grid points N to be used for the finite difference method.

• Calculate the discrete location of the points on the grid:

•  Calculate the spacing between the grid points 

•  Use the finite difference technique to represent the 2^nd order PDE’s as ODE’s

 

•  Use the MOL to convert the PDE’s to ODE’s

•  Solve the ODE’s in the above step. The Rosenbrock technique with the Kaps-Rentrop GRK4A coefficients was employed in this case, with N= 40.

Plotted below are the 3D solutions of vectors u & v.