IMPLICIT 2-LEVEL FINITE-DIFFERENCE METHODS FOR THE 2-DIMENSIONAL DIFFUSION EQUATION

Two new two-level implicit finite-difference methods for solving the constant-coefficient two-dimensional diffusion equation are developed by including weights in the discretisation of the equation. Firstly, a finite-difference equation is found which uses the same computational molecule as the Crank-Nicolson equation, is unconditionally stable and second-order accurate, but is found to be more accurate for the numerical tests that were run. Then a higher-order method is developed by elimination of the largest terms from the truncation error obtained from the modified equivalent equation. The resulting unconditionally stable fourth-order method is found to be much more accurate than both of the second-order methods. This new method is more stable than the fourth-order explicit method developed by Noye and Hayman in [1], but it requires more CPU time to run when the same spatial grid is used.