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Explicit vs. implicit FDM: Could you please suggest a reference?
The context is the finite difference modeling (FDM) of the transient diffusion equation (the linear one: $\dfrac{\partial T}{\partial t} = \alpha \dfrac{\partial^2 T}{\partial x^2}$).
Two approaches are available for modeling the evolution of $T$ in time: (i) explicit and (ii) implicit (e.g., the Crank-Nicolson method).
It was obvious to me that the explicit approach has a local (or compact) support whereas the implicit approach has a global support.
However, with some simple Google searches (and browsing through some 10+ books I could lay my hands on), I could not find any prior paper/text to cite by way of a reference.
I feel sure that it must have appeared in some or the paper (or perhaps even in a text-book); it's just that I can't locate it.
So, here is a request: please suggest me a reference where this observation (about the local vs. global support of the solution) is noted explicitly. Thanks in advance.
Best,
--Ajit
[E&OE]
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Comments
You can try checking the book
You can try checking the book 'The Complex Variable Boundary Element Method in Engineering Analysis' page 48. There's something about FDM. But if you don't find what you are looking for you an consult with HEW.
Yes, I could.
Yes, I could.
That's not what I've had in the mind, though!
Sorry, you misread me.
--Ajit
[E&OE]
Trefethen - Finite Difference and Spectral Methods for Ordinary
See page 161 in Trefethen's unpublished on-line book. It clearly states that the domain of dependence (support) is infinite for implicit finite difference methods, but the domain of dependence (support) is finite for explicit finite difference methods for the heat equation.
https://people.maths.ox.ac.uk/trefethen/4all.pdf
However, this is undoubtedly not the first place in which this observation was made.
Re: support and explicit vs. implicit methods
Hi Ben,
Thanks a bunch! Trefethen's book had simply escaped my mind.
Anyway, even earlier, I had not read this book the way it should be; I had only ``dipped'' here and there into it, because at that time, I was doing PhD and had too many other things on my mind. Now that I am teaching an introductory course on CFD, it makes sense to read it cover-to-cover!
But let me tell you how delighted I was to read, on the same page, the passage which begins with: ``Between these two situations lies...''
I had a somewhat similar idea, and had in fact pursued my thinking along the lines like: the possibility that even in the linear diffusion, there could possibly be a variable speed of propagation for the diffusion ``front,'' where by ``front'' I meant, in 1D, the edges of the base of the CFL triangle. I had in fact wondered if there exists a theory wherein the dimensionality of a problem goes on increasing with time. You see, I had wondered if I can take the explicit solution (or its continuous analog), expand it into a suitable finite (or infinite) basis set (I meant ideas like function spaces), and see if a theory of increasing dimensionality had any neat theorem or so to offer for such formulations.
Anyway, thanks again!
--Ajit
[E&OE]
Your equation
I am finding in your listed equation that it has no problem to be solved with an explicit method which is a simple way. But if a second term (which doesn't appear) exists like a function the discretization may need an implicit method and the solving process will take more time.
Reply to Mohamed
Dear Mohamed,
If you mean to say that the diffusion equation as written in the main post above wouldn't require an implicit method because there is no source/sink term mentioned in it, and therefore it is better to go with an explicit method because the latter is simpler, then I have the following comment:
Well, the existence of a source/sink has no relevance in deciding which solution approach (explicit or implicit) to choose; other factors are, like: (i) stability considerations, (ii) computational costs for a given level of accuracy and the purpose of the calculation---whether you are solving the diffusion equation to model the time-evolution of a diffusion process itself, or as a quick iterative means (pseudo time-marching) for solving a steady (i.e. Poisson) problem).
Hope this helps.
--Ajit
[E&OE]
Reference FDM
Dear Ajit,
You are right the equation can be solved by an explicit scheme like FTCS, Richardson or 3 Level schemes. Another way is to use the unconditionally stable crank nicolson implicit scheme which uses the average of n-th and (n+1)-th time levels or 3 LFI scheme. This is well explained in Fletcher of Springer-Verlag as well as the condition on the chosen steps in the stability analysis.