I am trying to solve 3D wave equation expressed by stress-velocity formulation with Chebyshev spectral method. In the time domain, I am currently using RK4 explicit scheme but it is really slow. Any other ones work better? I had good experience with leap frog method but it is designed for 2nd order stress-displacement formulation. Thanks.
Hi Bo, from my experience, first order central difference formula works well for such problem. What are your boundary conditions? As I just added an answer to your question regarding PML you can try to implement implicit method on boundaries (as suggested in paper from Renaut and Frohlich https://www.google.dk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&ved=0CCcQFjAB&url=https%3A%2F%2Fmath.la.asu.edu%2F~rosie%2Fmypapers%2Frenautfrojcp.pdf&ei=sGzjU8n-EeLnyQOik4CADQ&usg=AFQjCNFAm3ec_AApZPKNsEAIX8ojAr8fTQ&bvm=bv.72676100,d.bGQ&cad=rja) which will allow you increase your time step.
I think the question that must be answered first is whether you really need 4th-order accuracy in time. Many simulations do not. If 2nd-order accuracy is acceptable, then there are many other options that will decrease your execution time. You mention leap-frog, which is 2nd-order, so I assume this is OK. However, leap-frog is only neutrally stable. Might be better to try Lax-Wendroff or MacCormack methods. If you don't already have a copy, get Computational Fluid Mechanics and Heat Transfer by Tannehill et al. Chapter 4 has a catalog that gives a good synopsis of numerous time integration methods. I'm certain you'll find useful information to help solve your problem.
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