I currently am using the Crank-Nicholsen method to solve the diffusion equation. I'm using the usual first order in time and second order in space but I have an application which I suspect requires to be more accurate in space and time. Are there any methods out there which offer increased accuracy? My only other solution is to have very refined meshes.
High order and high accuracy are distinct issues for finite differences, since mesh refinement-necessary for reducing the local error-implies doing more steps for any finite time, which increases the global error.
It might be a good idea to turn to (pseudo)spectral methods snd fast transforms.
Crank-Nicolson should be second order accurate in time. However, despite its popularity and its numerical stability, it does provide for an oscillatory evolution in time. Personally, I prefer to use BDF2 as the second-order-in-time scheme because any oscillations reduce rapidly in time.
I haven't yet tried compact differences for Fourier's equation. It may be a way to go? The thesis by Spotz might be what you're looking for:
https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.56.1164&rep=rep1&type=pdf
Amir Rezazadeh I'm not sure what you're asking here, I only really know about simple finite differences.
Stam Nicolis I'm working on a finite domain, would transform methods and pseudospectral methods work there?
D. Andrew S. Rees I downloaded the thesis, thank you for bringing this to my attention. The basic idea seems to include the higher-order truncations terms in the difference scheme and use the main equation to write high order derivatives in terms of lower-order ones.
Dear mat.
I will send you some examples with its coding. I hope be useful
Hi Mat Hunt !
Yes, that is correct. Thus far I have implemented these compact difference ideas solely for solving Poisson's equation in 2D and the accuracy is quite impressive.
However, I do wonder why your CN code doesn't work. Does an Euler method in time work?
This is exercise 7, which is related to keller box, which has an order 2 at the same time and place.
Open the code and change the part related to the initial conditions according to your problem if I understand what you mean.
Hi Mat! What precisely (equation and BC-wise) are you trying to solve? I find it difficult to accept that such systems cannot be solved numerically.
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