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Finite Difference for Laplace Operators

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Old   May 17, 2016, 07:16
Default Finite Difference for Laplace Operators
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I am working on the finite volume method on a clustered Cartesian grid for school. I am using a finite difference function I made previously to get the operators for u,v,d u/v d x/y for solving the Poisson equation.

The function evaluates the taylor expansion for each point of stencil x of phi about x0 and returns the coefficient matrix

[phi^0] \: \: \: \: \: \: \: [phi_i-1]
[phi^1] = A* [phi_i ]
[phi^2] \: \: \: \: \:\: \: [phi_i+1]

I am looking for the zeroth derivative of the phi as I am trying to use this to interpolate the value itself. How do I go about this? I have the values of phi on other points, but not on x0 itself.

Any help would be great
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Old   May 17, 2016, 11:06
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the zeroth derivative of phi in x0 is phi(i) ...

f(i+1) -f(i) = h* df/dx|i + h^2/2*d2f/dx^2|i + ...
f(i+1)-f(i) = -h* df/dx|i + h^2/2*d2f/dx^2|i

deltaf = A * derivatives + hot ->
derivatives = A^-1*deltaf - A^-1*hot

A is 2x2. Then, you disregard A^-1*hot and you get the vector derivatives_n which approximates the exact derivative

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