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I have a system of three coupled PDE and I discretized the equations using finite difference method.

It results in a block matrix equations as:

[A11 A12 A13] [x1] = [f1]

[A21 A22 A23] [x2] = [f2]

[A31 A32 A33] [x3] = [f3]

where, any of Aij is a square matrix.

I use segregated method to solve the system of equations iteratively as:

0 - initial values: x2=0, x3=0

1 - A11 * x1 = f1 - A12 * x2 - A13 * x3

2- update x1

3- A22 * x2 = f2 - A21 * x1 - A23 * x3

4- update x2

5- A33 * x3 = f3 - A31 * x1 - A32 * x2

6- update x3 and repeat 1 to 6 until convergence.

The issue I encountered is that when I change the order of the steps above I will get different results. For example if I first assume x1=x2=0 and I start by solving

A33 * x3 = f3 - A31 * x1 - A32 * x2 and continue with the other unknowns, I will get completely different results.

I hope you can help me to find out what is wrong in my method.

Thanks

Frank

p.s. I posted this initially in the Linear Algebra section but I got no reply but too many views. So, I thought it might be well suited for Differential Equation section.

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# Segregated method for numerical solution of a PDE system

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