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Old   February 17, 2005, 04:48
Default Looking for low-storage explicit time integration
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Recently 'Runge-Kutta' has encouraged us on this forum to start using (temporal) error control.

I'm willing to give it a go, but I require some extra properties to be provided by a time integration scheme.

I'm solving hyperbolic equations for aeroacoustics with a DG method, so memory demands are allready high. That's why I also want an explicit, low-storage scheme. Is someone aware of such a scheme?

Currently I'm using the low-storage 6-stage, 4th order RK, optimized for good dispersion and dissipation errors of Berland et al. (AIAA paper 2004-2814).
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Old   February 17, 2005, 14:55
Default Re: Looking for low-storage explicit time integrat
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There are a few papers on low-storage ERK methods. The topic was very important back in the days of the CRAY XMP-48 but is less so today. Most ERKs were designed thinking the user was not living at the stability boundary. Generally, this is where CFD lives.

Limiting that to ones with embedded methods narrows the field. I think that these do

but there may be some more recent ones. Once you're using a 3(2) or 4(3) pair, you're issue won't be accuracy but will be stability. The trick is to use a method and a controller that can comfortably live at the stability boundary. What you want is a controller that is at least SC-stable on the stability boundary.

"Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer Series in Comput. Mathematics, Vol. 14, Springer-Verlag 1991, Second revised edition 1996."

If a controller is SC-stable with a method then all real eigenvalues of the governing equations won't bother the controller. Imaginary ones my cause the step size to oscillate a bit from step to step. These constructs are not perfect but will give you some idea of what is going on. Also, since you will likely be very stability bound, your error will be near 10^(-6), or six decimal points of accuracy per step. This means that there is little benefit to methods whose dispersion or dissipation error is small - all temporal error is really small already.
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