It's known that we can approximate a solution of parabolic equations by replacing the equations with a finite difference equation. Namely, we can solve parabolic equations by Difference Equation Replacement or Crank-Nicolson methods. I need Matlab code of Crank-Nicolson method for attached problem
As a start I can recommend works from economics field
http://www.goddardconsulting.ca/matlab-finite-diff-crank-nicolson.html
Otherwise, there are plenty of Matlab implementations of the finite difference heat equation (physics).
Hope that helps, yours
Marcus
Crank Nicolson is a useful first tool, but I suggest you rather use the TR-BDF2 method
Hosea M, Shampine L. 1996. Analysis and implementation of TR-BDF2. Appl.
Numer. Math. 20: 21–37
which is equally accurate and efficient but far more robust. It is implemented in the native MATLAB solver ode23tb.
Dear Dr. Luca Bonaventura
Thank you for valuable suggestion. Strictly speaking I don't familiar with indicated method. But I think that it was helpful for me.
I think it is relevant to this discussion to point out that while Crank-Nicolson is second order in both space and time, one must be aware of non-physical oscillatory solutions that can arise in certain PDEs if too large a time step is taken.
I have generally found that a first order Backward-Euler discretization in time along with a second order central difference in space is a more robust alternative. Of course you lose second order convergence in time.
In Matlab, package, search for, Crank-Nicolson, there are some examples, which can help you.
I know the coding of the said method in Maple. If you need it please let me know.
Muhammad Usman Do you know the coding of Laplace's equation with mixed boundary condition in Maple?
i write a code for the special case. if you want it ,please let me know.
i write a Matlab code for the Modified Boussinesq equations in shallow water waves. there are getting some nonlinear terms, after using the FTSC scheme with Crank Nicholson method in discritization form. How to write Matlab code for Boussinesq equations. please help.
Md Uddin : If your parabolic PDE is nonlinear, the easiest and quickest approach is to use a forward difference method in time, the forward Euler discretisation of the time derivative term.
If you wish to solve your system using an implicit scheme, such as Backward Euler, Crank Nicholson, BDF2 and so, on then the single TDMA solve at each time step will be replaced by something like a Newton-Raphson iteration scheme for each time step. Therefore the first step in creating a numerical code will be to find out how the Newton-Raphson method may be applied to the PDE itself. Then that PDE is discretised, and a TDMA solve is performed for each iteration.
The above assumes that your PDE has only one independent variable in space. Similar ideas may also be used in 2D. Whatever code you would normally use for a 2D Fourier's equation may be modified by a suitable iteration loop within each time step.
Go to help section, if do a search for Crank Nicolson. You will find the way.
Dear Elvin Azizbayov
If you need the matlab code for CN scheme of special type of parabolic heat equation I am happy to help.
Best Regards
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