Hey all, I've been working with an in-house code for a while and have decided to pursue a different method of solving the pde inside. Currently a spatial centered difference scheme is used to give us our equation to solve with the Jacobi iterative method. I want to investigate the conjugate gradient method as it pertains to the problem, but have been struggling. I need a new set of eyes.(adsbygoogle = window.adsbygoogle || []).push({});

So first off, here is the equation inside the Jacobi: (Let i denote x direction, j the y direction)

[itex]\Psi_{i,j} = \kappa_A \left(\left(\Delta x\right)^2 * \left(x_{i,j}^2 + y_{i,j}^2\right) * \omega_{i,j} + \Psi_{i+1,j} + \Psi_{i-1,j} + \kappa_2*\left(\Psi_{i,j+1} + \Psi_{i,j-1}\right)\right)[/itex]

Where [itex]\Delta x[/itex], [itex]\kappa_A[/itex], and [itex]\kappa_2[/itex] are scalar constants.

[itex]\Psi[/itex] is what I'm solving for, [itex]\omega[/itex] is the RHS.

So for conjugate gradient method (CGM) we need to rearrange this to the form Ax = b, where A is an NxN symmetric positive definite matrix. So this is where I need someone to double check my work. I started with the above equation and rearranged it to:

[itex]\Psi_{i,j}/\kappa_A - \Psi_{i+1,j} - \Psi_{i-1,j} - \kappa_2*\left(\Psi_{i,j+1} + \Psi_{i,j-1}\right) = \left(\Delta x\right)^2\left(x_{i,j}^2 + y_{i,j}^2\right)*\omega_{i,j}[/itex]

Assuming that is correct, then the matrix A should consist of [itex]1/\kappa_A[/itex] along the diagonal, -1's on both sides of the diagonal and then [itex]-\kappa_2[/itex] at each side spaced out from the diagonal based on the size of [itex]\Psi[/itex].

Might look something like this:

The vector, x, would then be [itex] \left[ \Psi_{1:end,1};\Psi_{1:end,2};... \Psi_{1:end,end}\right] [/itex]

And vector, b, would be [itex]\left(\Delta x\right)^2 * \left[\left(x_{1:end,1}^2 + y_{1:end,1}^2\right)*\omega_{1:end,1};... \left(x_{1:end,end}^2 + y_{1:end,end}^2\right)*\omega_{1:end,end}\right] [/itex]

So how have I done? Am I good up to here?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Conversion from Centered Diff Scheme to Ax = b

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**