What is the simplest, fastest and best method for solving moving boundary problem?
In general case this is complicated and the best approach may depend on particular problem. For example, for waves on the surface of water we have the system of equations including one on the moving surface (see eq. (3-4) in the attached article). If deviation is rather small, we can use expansion in series of powers of small parameter (5) to derive the set of equations that include this deviation from zero as one of the functions. Further (in this model) a modification of KdV equation (nonlinear PDE (9)) is derived.
https://www.researchgate.net/publication/305641177_Interaction_of_a_Solitary_Tsunami_Wave_with_River_Current
If you are interested more in numerical schemes, you should also use similar methods to derive a difference scheme with which one can work. But in the case considered above we do not know the movement law for a boundary and have to derive it. If it is known, probably numerical method is simpler.The technical problem will be to adjust your grid at every time step so that it covers the boundary.
Conference Paper Interaction of a Solitary Tsunami Wave with River Current
There are many the keywords are: Level Set, Lagrangian methods, Volume of fluid. They have different characteristics depending on the degree of deformation.
Dear Abdorrazzagh,
As usual, there is nothing like "the best method". Whether one ore another method is goed and how efficient it is compared to other methods depends on the problem itself. So I think it would help if you provide more details in this sense.
Of course, methods like the ones mentioned before are good candidates. I would add maybe the ALE method. But, again, I think it is better if you first analyze the problem and after this you decide for one method or another... or maybe even find your own.
Good luck,
Sorin
I found something relevant on Arxiv today. http://arxiv.org/pdf/1609.03373.pdf
It depends on the type of problems. However, FD for solving both linear and nonlinear set of equations can be used.
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