central difference approximation

Hermano · January 4, 2011, 08:49

Hi,

Where can I find the expression of the central difference approximation of the first and second derrivative (spatial) for a NON uniform grid?

otd · January 4, 2011, 18:00

For 3 points, (x1, f1), (x2, f2), (x3, f3), x1 < x2 <x3

Two methods (identical results for the approximation):

1. Expand f1 and f3 about x2 in Taylor series. Expand through the second derivatives.

You get two equations with unknowns df/dx and d^2f/dx^2 evaluated at x2. Solve those for your solution. Note: Expand through the 3rd and 4th derivatives, you have terms for the major errors you ignore by truncating the series after the 2nd derivatives.

2. Fit a parabola, f(x) = f2 + b(x - x2) + c(x - x2)^2 through the 3 points. b and c will be functions of x1, x2, x3 and f2 and f3.

Good luck.

OTD

DoHander · January 5, 2011, 17:23

Try this book:

Ferziger and Peric - Computational Methods for Fluid Dynamics

Do

January 4, 2011, 08:49	central difference approximation	#1
Hermano New Member Join Date: Jul 2010 Posts: 10 Rep Power: 15	Hi, Where can I find the expression of the central difference approximation of the first and second derrivative (spatial) for a NON uniform grid?

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January 4, 2011, 18:00		#2
otd Member private Join Date: Mar 2009 Posts: 74 Rep Power: 17	For 3 points, (x1, f1), (x2, f2), (x3, f3), x1 < x2 <x3 Two methods (identical results for the approximation): 1. Expand f1 and f3 about x2 in Taylor series. Expand through the second derivatives. You get two equations with unknowns df/dx and d^2f/dx^2 evaluated at x2. Solve those for your solution. Note: Expand through the 3rd and 4th derivatives, you have terms for the major errors you ignore by truncating the series after the 2nd derivatives. 2. Fit a parabola, f(x) = f2 + b(x - x2) + c(x - x2)^2 through the 3 points. b and c will be functions of x1, x2, x3 and f2 and f3. Good luck. OTD

January 5, 2011, 17:23		#3
DoHander Senior Member Join Date: Nov 2009 Posts: 411 Rep Power: 19	Try this book: Ferziger and Peric - Computational Methods for Fluid Dynamics Do