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I have an advection problem with a sever time stepping restriction. I am using explicit RK type time stepping schemes and standard finite difference so everything is matrix free. Other than going for an implicit scheme, are there some purely numerical schemes I can look into to alleviate my time stepping restriction and perhaps accelerate my time to solution?

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    $\begingroup$ Can you give some more details on your problem? Is it linear/nonlinear, size of the discretized system, etc. For example, if you have a linear problem with very large eigenvalues, there might not be much you can do to improve explicit time-stepping $\endgroup$
    – whpowell96
    Commented May 10 at 15:32
  • $\begingroup$ Yeah I think I got my answer $\endgroup$ Commented May 10 at 15:34
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    $\begingroup$ For strictly hyperbolic PDEs your options are somewhat limited. The best you usually can do is pick a different RK method which has a higher effective CFL per stage. If you have something mixed hyperbolic/parabolic and its the parabolic part which is causing problems you can try Runge-Kutta-Chebychev methods. $\endgroup$ Commented May 10 at 15:59
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    $\begingroup$ There are also mixed implicit-explicit time stepping methods where you solve the stiff parts implicitly and everything else explicitly, but this might not be what you're looking for and it's by no means a panacea. $\endgroup$ Commented May 10 at 16:01

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