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I have a system of 256 differential equations that I want to solve numerically. The system represents the Liouville equation, which is a first order, linear differential equation with complex numbers.
What's the best method to solve this equation in parallel? Please advise.
I'm a physicist, so I'm familiar with the basic solvers (Euler and Runge-Kutta), which are basically sequential by definition.
256 equations is a relatively small number. All of the usual integrators, such as those included in Matlab, Maple or Mathematica should have no real problem with equations of this size and should be able to return answers in a fraction of the time it would take an algorithm you would implement yourself, because they use sophisticated explicit/implicit and adaptive time stepping methods.
ODE solvers are, for the most part, a solved problem. Don't waste your time implementing one yourself -- just use what others have done before you.
I would disagree with your statement that Euler and Runge-Kutta have to be sequential. I know that you are saying this because you cannot parallelize across time steps (or across inner time steps in the case of something like RK4), however both can be parallel as long as you do so within a single time step. You would simply evaluate the derivatives of your 256 variables in parallel.
Depending on your system characteristics, it is possible that RK methods would be the best for you, or perhaps a simple stiff solver. I personally am not familiar with parallel implementations so I will leave the recommendations to others with much more experience than myself. Unless your system is something incredibly special though, you will be using algorithms that you are more than likely familiar with.
From the comments, it sounds like your problem is stiff, and using an implicit integrator will help you a lot more than trying to parallelize. But in case someone comes here looking for information on parallel time integration, you can find a discussion of some simple parallel extrapolation and deferred correction methods in this preprint of mine. It's not a new topic; you can trace the references backward starting from the bibliography there. For more sophisticated methods, see answers to this question.
Parallelism in time is possible, but is only useful if you need high accuracy (relative to what a cheap stable integrator will give you).
You may want to look at parareal methods, that is, a parallel-in-time approach. There can be huge gains in speedup, but there are some issues with stability. The paper by Lions, Maday & Turinici (2001) would be a good place to start if you're not familiar with the method.
