Finite Difference for Laplace Operators

Snow360 · May 17, 2016, 07:16

Hi,

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

FMDenaro · May 17, 2016, 11:06

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

May 17, 2016, 07:16	Finite Difference for Laplace Operators	#1
Snow360 New Member Join Date: Feb 2016 Posts: 14 Rep Power: 10	Hi, 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

Thread Tools	Search this Thread
Show Printable Version Email this Page	Search this Thread: Advanced Search
Display Modes
Linear Mode Switch to Hybrid Mode Switch to Threaded Mode

Similar Threads
Thread	Thread Starter	Forum	Replies	Last Post
The fundamental difference between Finite Difference and Finite Volume Methods	vikramaditya91	Main CFD Forum	3	January 14, 2016 04:13
Finite element vs. finite difference	Francisco Saldarriaga	Main CFD Forum	23	December 17, 2014 08:21
Finite difference method	kaliramesh	FLUENT	1	January 16, 2014 12:55
Finite difference can be non-conservative?	gregmason	Main CFD Forum	2	July 8, 2009 11:53
Fininte difference and Finite element Technique	Mahendra Singh Mehra	FLUENT	3	December 22, 2005 23:49

May 17, 2016, 11:06		#2
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,763 Rep Power: 71	the zeroth derivative of phi in x0 is phi(i) ... f(i+1) -f(i) = h* df/dx\|i + h^2/2d2f/dx^2\|i + ... f(i+1)-f(i) = -h df/dx\|i + h^2/2d2f/dx^2\|i deltaf* = *A derivatives + hot -> derivatives = A^-1deltaf - A^-1hot A is 2x2. Then, you disregard A^-1hot* and you get the vector derivatives_n which approximates the exact derivative