To solve the heat conduction (or diffusion) equation, it is well known that explicit schemes (like FTCS) are rarely used, because very tiny timesteps are required to avoid instability. I know that implicit methods usually have no such problem, but they have to handle huge matrices and cannot be trivially parallelized.
However, there are explicit methods with much better stability properties:
- Runge–Kutta–Chebyshev,
- ADE (Alternating Direction Explicit),
- Hopscotch,
- Dufort-Frankel (etc.?)
Now I can not really understand why they haven't become widespread in the practice. Have they got serious disadvantages? If yes, what are them? Or maybe only because people are unaware of them?
I think you can find useful the reading of the the first chapters in this book where parabolic equations are discussed
Morton K., Mayers D. Numerical solution of partial differential equations. An introduction (2ed., CUP, 2005)(ISBN 0521607930)(293s)
you will find several schemes that are presented and anlysed in terms of accuracy and stability.
In my experience, the parabolic part is only one of a more complex mathematical system of PDEs. Working on flows like boundary-layer type using non-uniform grids the stability constraint in explicit methods is a big problem.
The answer to your question goes as follows.
When you discretize a PDE, you essentially replace the PDE with a system of ODEs - your numerical scheme solves the ODE system. This is not apparent when you implement the scheme, but this is what the scheme is doing.
That system of ODEs is characterized by a large spectrum of eigenvalues, many of which are relevant to the physics of the problem you are solving, and many more of which are purely a numerical artifact and are not needed to capture the physics of your problem. In an explicit scheme, the time step is constrained by the "irrelevant" portion of the eigenvalue spectrum. In other words, when you use an explicit scheme, your time step is limited by something that doesn't matter to your solution. With an implicit scheme, on the other hand, the time step is constrained, not by stability, but by the need to accurately capture the physics represented in the most extreme eigenvalues within the "relevant" part of the spectrum.
The question is why explicit schemes are not used and then you cite
- R K Chebychev method
- ADE
- Hopscotch
- Dufort-Frankel scheme
There are three properties must be discussed about a scheme which are consistency, convergence and stability. If a scheme is consistent then stability implies convergence.
Let us considered du Fort - Frankel scheme, it is three level scheme which is unconditionally stable but not consistent unless time step k is smaller than space step h (k must tend to zero faster than h,i.e. k
Dear professor R. C. Mittal,
Thank you for your answer!
I'm talking about ADE methods, like Özisik in his book Finite Difference
Methods in Heat Transfer, 6.4. ("Saul’yev (1957) was the first to propose a two-step scheme, which is unconditionally stable ... Later on, Larkin (1964) and Barakat and Clark (1966) proposed alternative ADE schemes..." etc.)
Anyway, I would be grateful if you could suggest to me (downloadable) literature where these methods are discussed and their limitations are demonstrated in practical cases.
I think you can find the response in Remark 8.7 pp267 in this book:
H. Le Dret, B. Lucquin, Partial Differential Equations: Modeling Analysis and Numerical Approximation, vol 168, Springer International Publishing Switzerland 2016.
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