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→The method: The infinitesimal change in x and y didn't appear in the equations! see Press 1992 Numerical Recipes in C p 855-856 |
→The method: Continuation of the previous change for the split equations |
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with:
: <math>\Delta = \Delta x = \Delta y</math>
where <math>\delta_p</math> is the central difference operator for the ''p''-coordinate. After performing a [[Von Neumann stability analysis|stability analysis]], it can be shown that this method will be stable for any <math>\Delta t</math>.
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: <math>{u_{ij}^{n+1/2}-u_{ij}^n\over \Delta t/2} =
\left(\delta_x^2 u_{ij}^{n+1/2}+\delta_y^2 u_{ij}^{n}\right)\over \Delta</math>
: <math>{u_{ij}^{n+1}-u_{ij}^{n+1/2}\over \Delta t/2} =
\left(\delta_x^2 u_{ij}^{n+1/2}+\delta_y^2 u_{ij}^{n+1}\right)
The system of equations involved is [[symmetric matrix|symmetric]] and tridiagonal (banded with bandwidth 3), and is typically solved using [[tridiagonal matrix algorithm]].
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