Difference between revisions of "Numerical Differentiation"

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| Backward Difference
 
| Backward Difference
| <math> \left. \frac{\mathrm{d}}{\mathrm{d}x} \right|_{i} \approx \frac{\phi_{i}-\phi_{i-1}}{\Delta x}</math>
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| <math> \left. \frac{\mathrm{d}\phi}{\mathrm{d}x} \right|_{i} \approx \frac{\phi_{i}-\phi_{i-1}}{\Delta x}</math>
 
| <math>\mathcal{O}\left(\Delta x \right) </math>
 
| <math>\mathcal{O}\left(\Delta x \right) </math>
 
|-
 
|-
 
| Central Difference
 
| Central Difference
| | <math> \left. \frac{\mathrm{d}}{\mathrm{d}x} \right|_{i} \approx\frac{\mathrm{d}}{\mathrm{d}x} \approx \frac{\phi_{i+1}-\phi_{i-1}}{2\Delta x}</math>
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| | <math> \left. \frac{\mathrm{d\phi}}{\mathrm{d}x} \right|_{i} \approx \frac{\phi_{i+1}-\phi_{i-1}}{2\Delta x}</math>
 
| <math>\mathcal{O}\left(\Delta x^2 \right) </math>
 
| <math>\mathcal{O}\left(\Delta x^2 \right) </math>
 
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Revision as of 09:24, 7 July 2008

Introduction

Taylor Series

Lagrange Polynomials

Lagrange polynomials, which are commonly used for interpolation, can also be used for differentiation.



Tables for Derivatives on Uniform Grids

Some First Derivative Expressions for a Uniform Mesh
Derivative at point i Discrete Representation (uniform mesh) Order
Forward Difference  \left. \frac{\mathrm{d}\phi}{\mathrm{d}x} \right|_{i} \approx \frac{\phi_{i+1}-\phi_{i}}{\Delta x} \mathcal{O}\left(\Delta x \right)
Backward Difference  \left. \frac{\mathrm{d}\phi}{\mathrm{d}x} \right|_{i} \approx \frac{\phi_{i}-\phi_{i-1}}{\Delta x} \mathcal{O}\left(\Delta x \right)
Central Difference  \left. \frac{\mathrm{d\phi}}{\mathrm{d}x} \right|_{i} \approx \frac{\phi_{i+1}-\phi_{i-1}}{2\Delta x} \mathcal{O}\left(\Delta x^2 \right)


Some Second Derivative Expressions for a Uniform Mesh
Derivative at point i Discrete Representation (uniform mesh) Order