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As part of my research I came up with a boundary value problem where I need to solve the following system of coupled PDE:

1- a1 * f,xx + a2 * f,yy + a3 * g,xx + a4 * g,yy - a5 * f - a6 * g = 0

2- b1 * f,xx + b2 * f,yy + b3 * g,xx + b4 * g,yy - b5 * f - b6 * g = 0

Where, ai's and bi's are all positive constants.

To solve this system numerically, I discretize it using finite difference scheme. In Eq. 1, I move all the terms associated with g(x,y) to the RHS and I solve Eq.1 for f(x,y).

Then, in Eq.2 I move all terms associated with f(x,y) (that is updated from the previous step) to RHS and I solve Eq.2 for g(x,y) and I continue this process until f and g converge to a unique solution.

I am wondering if there exists a more efficient way than my method to solve this system numerically and using finite difference scheme.

Your comments are highly appreciated.

Frank