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?"

Comments

Be much more specific...

[gowen]

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).