There are known finite difference approximations for derivatives of first few orders. They are in three groups. Forward, Backward and central.
1. Forward difference gives derivative at the leftmost of points involved in the formula.
2. Backward difference gives derivative at the rightmost of points involved in the formula.
3. central difference gives derivative at the center of points involved in the formula.
I want to know of any known finite difference formula (of any convergence order) that gives derivative(of any order) at any of the points involved.
Dear Nasir, it seems to me there are many more schemes especially different combinations of the three groups you have mentioned. Maybe, as a good reference you can have a look at
Strikwerda, J.C. (1989), Finite Difference Schemes and Partial Differential Equations, Wadsworth & Books/Cole, USA.
And regarding the convergence order, there exist also acceleration methods by using which you can increase the order of convergence, e.g. Richardson extrapolation; Richardson extrapolation is very well known and you can simply find many references for it. With best wishes, have a nice future ahead.
See appendix A of my book "Numerical Calculus" http://www.amazon.com/dp/B07BS1DN1S it won't be available for free again until 9/25, 10/3, 10/11. The software (in particular source code to generate a wide variety of difference equations) is always free at http://dudleybenton.altervista.org/software/index.html There's a lot of other software (and source code) there too. You can always send me an Email.
Please go through the following book:
Carnahan B., Luther H.A. and Wilkes J.O., 1969 “Applied Numerical Methods”, John Wiley and sons, New York.
Yes there are! two issues you have to consider
1- We can have 3rd order, 4th order or exponential order (like spectral method) diffrentiation ..., you can use pade approximation and then get the analytical derivative of the function, you can have higher order with Compact Difference Method (it is sometime called method of differed correction), this concept can expanded to non equal distance locations for numerical differentiation and it is called " Differential quadrature"... Any book on numerical methods has a section on numerical differentiation.
2- But, not all of the methods are developed for the sake of the order of accuracy of differentiation method. There are other concerned in numerical differentiation like stability and wiggles when the method is being used for building a response surface. There are thorough literature on those issues of numerical differentiation as well, numerical smoothing ... so do not think only order of accuracy is the concern in numerical differentiation, it is sometime the secondary issue
Thank you all for the answers. I have found an online calculator to generate finite difference formulas as I wanted.
It can be found in
http://web.media.mit.edu/~crtaylor/calculator.html
Thanks
Nasir
It also depends upon the context. If you need to differentiate nonlinear terms then there are multiple choices: for example, should you code 2uu_x or (u^2)_x or some linear combination of these? or something else? Extra principles need to decide. For general nonlinear PDEs Centre Manifold Theory provides a systematic rational for deciding which of the derivatives to use. See Chapter 15 in my book (2015), Model emergent dynamics in complex systems, SIAM, isbn: 9781611973556, http://bookstore.siam.org/mm20/ or articles such asArticle Holistic discretization ensures fidelity to Burgers' equation
Article A dynamical systems approach to simulating macroscale spatia...
and see the following web service for discretisation of PDEs in one space dimensionhttp://www.maths.adelaide.edu.au/anthony.roberts/holistic1.php
using taylor series you can find several formulas with higher order approximations
is there any method derive the following formula
Forward Difference Formula for second derivative of accuracy O(h2)
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