Parallel implementation of a high-order implicit collocation method for the heat equation

We combine a high-order compact finite difference approximation and collocation techniques to numerically solve the two-dimensional heat equation. The resulting method is implicit and can be parallelized with a strategy that allows parallelization across both time and space. We compare the parallel implementation of the new method with a classical implicit method, namely the Crank–Nicolson method, where the parallelization is done across space only. We find the set of conditions for which each method is more advantageous than the other. Numerical experiments are carried out on the SGI Origin 2000.