Fortran Programming Language :: How Accurate is Runge-Kutta?


	How Accurate is Runge-Kutta? Hello. I'm currently investigating some strange results I'm getting in a simulation using the fourth Order Runge-Kutta algorithm. Basically I'm quite certain that the numerical method is to blame for some slightly errornous results cropping up in the more extreme regions of my simulation. However, I have just tried halving the stepsize used, and it made no difference. Is it possible that there are inhenrent inaccuracies in the Fourth Order Runge-Kutta method that no matter how low the stepsize goes, they will always be there? Kind Regards, Matt Matt wrote: > Hello. I'm currently investigating some strange results I'm getting in > a simulation using the fourth Order Runge-Kutta algorithm. Basically > I'm quite certain that the numerical method is to blame for some > slightly errornous results cropping up in the more extreme regions of > my simulation. However, I have just tried halving the stepsize used, > and it made no difference. > Is it possible that there are inhenrent inaccuracies in the Fourth > Order Runge-Kutta method that no matter how low the stepsize goes, > they will always be there? Yes. Runge-Kutta methods, and most other methods of this sort, assume that the equations that they are solving have continuous solutions. If the true solution is discontinuous, or has discontinuous derivatives, then you will find errors that do not decrease with the expected order of accuracy of the method, and in some cases you will find errors that do not decrease at all. Furthermore, the convergence theory regarding Runge-Kutta methods assumes perfect arithmetic. This is not true inside a computer, of course. If these "extreme regions" of your simulation involve large coefficients that nearly cancel, for example, the numerical roundoff errors can be quite significant. Similarly, if you have a very tiny step size, you can also get roundoff errors that accumulate to significant levels. However, if the solution to your equations does, in fact, meet the necessary continuity requirements, and there are no problems due to numerical roundoff error, and there are no errors from other portions of the calculation, then it is fundamentally true that in the limit as the Runge-Kutta step size is decreased the errors will converge to zero. (Er, right -- this also assumes that the method is in fact stable for the relevant equation. I believe that the standard coefficients for the fourth-order Runge Kutta scheme are stable for very tiny step sizes regardless of the differential equation involved, but I wouldn't swear to it, and I presume there are also nonstandard coefficient choices that aren't stable.) - Brooks -- The "bmoses-nospam" address is valid; no unmunging needed. "Matt" <matt@hotmail.com> wrote in message news:1177624455.861827.12070@o40g2000prh.googlegroups.com... > Hello. I'm currently investigating some strange results I'm getting in > a simulation using the fourth Order Runge-Kutta algorithm. Basically > I'm quite certain that the numerical method is to blame for some > slightly errornous results cropping up in the more extreme regions of > my simulation. However, I have just tried halving the stepsize used, > and it made no difference. > Is it possible that there are inhenrent inaccuracies in the Fourth > Order Runge-Kutta method that no matter how low the stepsize goes, > they will always be there? > Kind Regards, Integral calculus is a misnomer. If it were, we might understand how Dick Feynman operated. What are you integrating and over what space? -- WW The 1961 "Modern Computing Methods" HM Stationary Office notes for the Runge-Kutta method (Chapter 9) that the main disadvantage is instability The problem arises in that the finite difference equation upon which the method is based may have an unwanted increasing solution where the underlying original set of equations does not. The only hope offered is to use smaller and smaller steps until a different behaviour is found. More reasons are given in section 27 of chapter 9. Further methods suggested are that of De Vogelaere and the use of Chebyshev series. On Apr 26, 10:54 pm, Matt <matt@hotmail.com> wrote: > Hello. I'm currently investigating some strange results I'm getting in > a simulation using the fourth Order Runge-Kutta algorithm. Basically > I'm quite certain that the numerical method is to blame for some > slightly errornous results cropping up in the more extreme regions of > my simulation. However, I have just tried halving the stepsize used, > and it made no difference. > Is it possible that there are inhenrent inaccuracies in the Fourth > Order Runge-Kutta method that no matter how low the stepsize goes, > they will always be there? Check your code first. Take a system with known solution, then check how the accuracy of your code changes with step size. Easy enough to get one of the constants wrong. Halving the step size won't make a difference once you are at the limit of your floating-point arithmetic. You are not using "float" by any chance? Do you have any step control based on error estimates? You will always have problems if the function involved doesn't have fourth derivatives (maybe fifth), or something like sqrt (x) close to 0. And what exactly do you mean by "it made no difference"? Terence wrote: > The 1961 "Modern Computing Methods" HM Stationary Office notes for the > Runge-Kutta method (Chapter 9) that the main disadvantage is > instability The problem arises in that the finite difference equation > upon which the method is based may have an unwanted increasing > solution where the underlying original set of equations does not. I believe this is true for all initial value that integrate the solution in the direction of an undesired exponentially growing solution. It results in diverging solutions. If the OP didn't go too far, it might be just enough of a difference to seem wrong, before it really diverges. -- glen On Apr 26, 2:54 pm, Matt <matt@hotmail.com> wrote: > Hello. I'm currently investigating some strange results I'm getting in > a simulation using the fourth Order Runge-Kutta algorithm. Basically > I'm quite certain that the numerical method is to blame for some > slightly errornous results cropping up in the more extreme regions of > my simulation. However, I have just tried halving the stepsize used, > and it made no difference. > Is it possible that there are inhenrent inaccuracies in the Fourth > Order Runge-Kutta method that no matter how low the stepsize goes, > they will always be there? If your system of DE's is "stiff", you may need to go with another type of integrator, such as one of the predictor-corrector methods. Runge-Kutta simply does not always work. There are whole research groups in places like Electrical Engineering or Physics departments who work, even today, at developing better numerical DE solvers, and they do so because they have to: for many circuit-design or power system problems, classical methods like Runge-Kutta are just not good enough. Choosing a best step-size can be a non-trivial problem: if it is too large, truncation errors are excessive, but if it is too small, roundoff errors ruin everything. If you want really small step-sizes, you may need to go with extended-precision arithmetic. Most numerical analysis books or books about the numerical solution of DEs will contain discussions of stepsize choice. Good luck. R.G. Vickson - Hide quoted text - > Kind Regards, > Matt On Apr 26, 3:54 pm, Matt <matt@hotmail.com> wrote: > Hello. I'm currently investigating some strange results I'm getting in > a simulation using the fourth Order Runge-Kutta algorithm. Basically > I'm quite certain that the numerical method is to blame for some > slightly errornous results cropping up in the more extreme regions of > my simulation. However, I have just tried halving the stepsize used, > and it made no difference. > Is it possible that there are inhenrent inaccuracies in the Fourth > Order Runge-Kutta method that no matter how low the stepsize goes, > they will always be there? > Kind Regards, > Matt Any explicit RK method of order higher than 2 may converge to spurious steady-state solutions if run with fixed time stepsize. Changing the stepsize during the run sometimes helps, but running with another fixed stepsize doesnt. If you suspect this is the problem, force the integrator to vary the step randomly (but within tight bounds) and see what happens. The classical reference for this phenomenom, which is common in long-term simulations of gradient (Hamiltonian) systems (& also in chaotic systems) is J. M. Sanz Serna, Numerical ordinary differential equations versus dynamical systems, in "The dynamics of numerics and the numerics of dynamics", ed by Broomhead and Iserles, Cambridge, 1992. - Hide quoted text - Matt wrote: > Hello. I'm currently investigating some strange results I'm getting in > a simulation using the fourth Order Runge-Kutta algorithm. Basically > I'm quite certain that the numerical method is to blame for some > slightly errornous results cropping up in the more extreme regions of > my simulation. However, I have just tried halving the stepsize used, > and it made no difference. > Is it possible that there are inhenrent inaccuracies in the Fourth > Order Runge-Kutta method that no matter how low the stepsize goes, > they will always be there? > Kind Regards, > Matt If you suspect your method, why not pick up a tested code at netlib.org. I happen to have a preference for the codes on my site at http://mathalacarte.com/. And yes if the stepsize goes to 0, and rounding errors are not a problem, the error should get better if you don't have other mistakes. Halving the stepsize may not have been nearly a large enough reduction. Fred On Apr 27, 3:51 pm, Fred Krogh <fkr@mathalacarte.com> wrote: - Hide quoted text - > Matt wrote: > > Hello. I'm currently investigating some strange results I'm getting in > > a simulation using the fourth Order Runge-Kutta algorithm. Basically > > I'm quite certain that the numerical method is to blame for some > > slightly errornous results cropping up in the more extreme regions of > > my simulation. However, I have just tried halving the stepsize used, > > and it made no difference. > > Is it possible that there are inhenrent inaccuracies in the Fourth > > Order Runge-Kutta method that no matter how low the stepsize goes, > > they will always be there? > > Kind Regards, > > Matt > If you suspect your method, why not pick up a tested code at netlib.org. > I happen to have a preference for the codes on my site athttp://mathalacarte.com/. > And yes if the stepsize goes to 0, and rounding errors are not a > problem, But, rounding errors are always a problem, at least in floating-point arithmetic. That's why the step-size choice problem is nontrivial. R.G. Vickson - Hide quoted text - > the error should get better if you don't have other mistakes. > Halving the stepsize may not have been nearly a large enough reduction. > Fred On 28 Apr 2007 17:37:23 -0700, "C@shaw.ca" <C@shaw.ca> wrote: >But, rounding errors are always a problem, at least in floating-point >arithmetic. That's why the step-size choice problem is nontrivial. No. Rounding is always an issue to keep in mind but it is not necessarily a problem.