Runge Kutta Optimization
Hello friends,
There are two directions that are pusued for a Runge Kutta Scheme. The first way is to increase the order (which also increases the number of stages of a Runge Kutta Scheme) to improve the solution from Runge Kutta scheme. The second way is to optimize the RK scheme(keep the order same and increase the number of stages) My question is why do we optimize it when we can increase its order. because the number of stages is still the same? Thanks in advance Vasanth 
Re: Runge Kutta Optimization
By keeping the order the same and increasing the number of stages it is sometimes possible to increase the stability bounds of the scheme allowing the use of larger time steps which in turn reduces the number of iterations required to convergence.
For time marching methods applied to steady problems the temporal order of accuracy is largely irrelevant and so it makes no sense to pursue higherorder of accuracy with RK schemes anyway. 
Re: Runge Kutta Optimization
First, RungeKutta methods include implicit types. The topic is far richer than just explicit methods.
Time steps are generally chosen based on one of the two limits: accuracy or stability. In most CFD applications, if you integrate with an explicit RungeKutta method (ERK) then you are stability bound. This implies that your accuracy is quite acceptable but you are living on the ragged edge of the stability domain. At this point you are concerned the ratio of the maximum step size divided by the work. The maximum stepsize is that which places your problem with its particular RHS eigenvalue structure on the linear stability boundary. The work is simply the number of stages of the ERK. Stability bound efficiency of ERKs is essentially the inviscid or viscous CFL divided by the number of stages. The trick then is to maximize this ratio subject to keeping the local integration error to an acceptable level. Maximal stability boundaries of ERKs have been studied by Abdulle (2002) for real eigenvalues http://portal.acm.org/citation.cfm?id=587153.587215 and by Kinnmark and Gray (1984,...) http://md1.csa.com/partners/viewreco...ecid=0878982CI for imaginary eigenvalues. The more important thing that routinely goes unaddressed is how to keep the integrator a tad off the stability boundary but never letting it go over. This requires a good error controller. Everytime you step over the boundary, the global error increases geometrically.Oops!! It all boils down to which integrator can reliably integrate your problem at some user specified error tolerance for the least effort. Of course, in order to do this, one has to actually know what error one is actually committing. 
Re: Runge Kutta Optimization
Thank you friends and special thanks to RungeKutta for that wornderful piece of info on ERK.I will go through these papers.

Re: Runge Kutta Optimization
What is the full journal name of the paper by Kinnmark & Gray? Are there any other papers on RK methods with increased stability region for hyperbolic systems?
How do the methods by Medovikov and Abdulle developed for parabolic systems compare to the similar methods developed by Someijer, Shampine, and Verwer (e.g. J. Comp. Appl. Math. 88, 1997, 315) Thanks for your insightfull expertise. Angen 
Re: Runge Kutta Optimization
http://www.sciencedirect.com/science...&_coverDate=04%2F30%2F1984&_cdi=5655&_orig=search&_st=13&_sort=d &view=c&_acct=C000050221&_version=1&_urlVersion=0& _userid=10&md5=5df33205a7bb604d6b33359e4a8f4173
I don't provide free literature searches!! Get off your %$^ and go to the library. Medovikov and Abdulle studied extended stability domains for problems with purely real eigenvalues. Kinnmark and company are concerned with the imaginary axis. 
Re: Runge Kutta Optimization
So how much do I need to pay for that? But let get serious.
I tried the link you provided in your last message but it returns an error. I tried a link in your previous message and I have got reference to Math. Comp. Simul. My library catalog does not return anything meaningful for Math. Comp. Simul. or any variations of it that are consistent with English. I know I can look for a book or a web site with references between abbreviations and full journal names. My second question was about comparison between methods of Medovikov and Abdulle on one side and the methods developed by Someirjer et. al. on the other side. Both groups developed methods for parabolic systems so the eigenvalues are real (or close to negative real axis) in both cases. I know I can implement all this methods, run extensive test cases and comparisons and than I will have an answer or I can get an answer by ordering this work to someone else and paying for it. However, I had impression that this forum is for a free exchange of information and not for marketing and sales. Angen 
All times are GMT 4. The time now is 16:53. 