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Numerically Approximating the Wave Equation?
ObsessiveMathsFreak writes "I'm an applied mathematician who has recently needed to obtain good numerical approximations to the classic second-order wave equation, preferably in three space dimensions. A lot of googling has not revealed much on what I had assumed would be a well-studied problem. Most of the standard numerical methods, finite difference/finite element methods, don't seem to work very well in the case of variable wave speed at different points in the domain, which is exactly the case that I need. Are any in this community working on numerically solving wave equation problems? What numerical methods do you use, and which programs do you find best suited to the task? How do you deal with stability issues, boundary/initial values, and other pitfalls? Are there different methods for electromagnetic wave problems? Finally, when the numbers have all been crunched, how do you visualize your hard-earned data?"
Not exactly my field, but if the issues with FE/FD are ones of stability and/or convergence, have you looked into using multigrid methods? It requires more memory, but a good preconditioned, multigrid method should help reduce problems of variable speeds, no?
and not make fun of this person for asking a question that has nothing to do with our hobby horses. I hope this question has a small number of posts. Only well-meaning and helpful ones.
Are you doing the time harmonic case (3-D Helmholtz) or an unsteady case?
What does the domain look like (regular/rectangular and you may be able to use spectral methods)? In irregular domains, multigrid methods seem to converge most quickly for elliptic equations, but again, that depends on their exact form.
You don't say what goes wrong with finite difference codes... For pure Adams-Bashforth schemes often give extremely good numerical stability. You talk about variable wave speeds, but the Mathworld equation you link to doesn't cover that. In many cases you can use multiple-scales/WKB approaches, but that depends on how the wave speed varies (relative to the wavelength).
Finally: there are many things for which Googling sucks. This is one. For an proper overview, try a proper textbook, like "Waves in Layered Media", mentioned above, or "Modern Methods in Analytical Acoustics" (Crighton, Dowling et al).
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Hi, I'm currently doing a Masters Thesis on Diffusion PDE's (Parabolic and Elliptic) correct me If i'm wrong but the wave equation is a hyperbolic PDE. I use a MATLAB Toolbox called PDE Toolbox GUI and although I don't use it I know it includes a hyperbolic solver. I haven't investigated how detailed it is, but I suggest having a look through the manual, to see if it meets your requirements. Cheers
I'm a plasma physicist and I work in the domain of radiofrequency waves propagation and absorption in fusion plasmas. I've been busy developping a code that solves the Maxwell's equations, which are equivalent to the wave equation (3-D full-wave calculation). The case of a plasma is tricky because it both time *and* space dispersive.
I won't be able to even start explaining this stuff in this post, but my code uses finite elements for the radial direction and Fourier decompositions for the two periodic directions of a fusion device. These numerical methods work well. I also know finite difference codes which work well. So, I think you should look a bit harder, because FE or FD methods usually do the trick, even for "variable wave speed at different points in the domain"... Regarding the boundary conditions, well, you'd better be very careful, because they will usually completely determine the solution. Again, it is my experience that Finite elements are well adapted to this task but you'll have to do some research.
Finally, for the vizualisation, matplotlib and vtk work for me.
First, try to determine and explain more precisely what it is you want to do: "to obtain good numerical approximations to the classic second-order wave equation, preferably in three space dimensions" sounds a but vague. Pick up the right textbooks, scientific journals, learn, exchange with the community. I know my post sounds a bit patronizing but this is science, and this kind of effort takes dedication, time and patience. I think Slashdot and Google are hardly the right places to start...
http://kim.oyhus.no/wave.html
It is a Java applet. Note the low dispersion. Try clicking on it!
In order to make this, I avoided the standard textbook methods.
They can give good waves, but at a high cost in computation and memory.
It is not standard finite differencing, since those methods introduce dispersion
and similar errors to an unnecessary degree. But it IS a finite differencing method,
and I have done variants of it with variable speed and in 3 dimensions.
Here is one with variable speed:
http://kim.oyhus.no/seismic.mpg
One of the tricks I use is to use a hexagonal grid.
Kim Øyhus, M.Sc. Physics