# Any questions about Runge-Kutta methods

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 January 26, 2005, 17:32 Any questions about Runge-Kutta methods #1 Runge_Kutta Guest   Posts: n/a Does anyone have any questions about Runge-Kutta methods? My motivation for posing this question is to see if I can find an employer who might be interested in hiring a fluid dynamicist/ numerical analyst. So, the deal is that I'll try to help you if you try to help me. OK?

 January 26, 2005, 18:16 Re: Any questions about Runge-Kutta methods #2 noName Guest   Posts: n/a Yes, I have one: Why do structural analysis people use their own set of numerical methods, like Newmark Beta, Wilson Theta, Collocation, Houbolt, Park etc? Why don't they use simple ode integration methods, like Runge Kutta Fehlberg?

 January 26, 2005, 18:32 Re: Any questions about Runge-Kutta methods #3 Runge_Kutta Guest   Posts: n/a Well, as I am not a structural analysis person, I can't really say much about this. Atmospheric science folks have a bunch of their own schemes also. What you tend to see is the mathematics community doing a serious analysis and physical scientists using what they understand. The net result is an enormous chasm between them. In my opinion, the fluid dynamics community is far better at spatial discretizations than temporal ones. Runge-Kutta methods come in many flavors. There are implicit, explicit, additive, and partitioned. One may then apply these to ordinary differential equations, differential algebraic equations, delay differential equations, etcetera. In fluid dynamics, we are generally concerned with ordinary differential equations and differential algebraic equations. As to your comment about RKF methods; much has happened since Fehlberg made those schemes in the late 1960's. Here are just a few things. It is generally accepted that the propagating solution should be the higher-order solution rather than the lower one as Fehlberg did. Next, people have devoted quite a bit of effort to minimizing the leading order truncation errors of new explicit RK pairs. One more thing is that error controllers have been systematically studied and PI controllers are much better than what Fehlberg used. If I could grab the ear of every numerical fluid dynamicist I would tell them to measure their temporal error and then adjust the time step to control it. Maybe 1% of people do that. Most people just close their eyes and put th gas pedal on the floor. Integration practices of most codes are astonishly primative.

 January 27, 2005, 04:50 Re: Any questions about Runge-Kutta methods #4 Yves Guest   Posts: n/a Well I have one question: For the integration of a Discontinuous Galerkin spatial discretization, is it necessary to have limiters whan using explict RK time integration? The model equations are Linearized Euler, but the coefficients of the mean flow. around which the linearization was done, can vary spatially. Your comments are most welcome. Yves

 January 27, 2005, 12:07 Re: Any questions about Runge-Kutta methods #5 Peter Attar Guest   Posts: n/a Well for one structural equations of motion are second order equations...so you would have to put them in first order form to use Runge-Kutta methods..which would entail increasing the storage of the arrays used in the computation by a factor of two.

 January 27, 2005, 14:17 Re: Any questions about Runge-Kutta methods #6 Runge_Kutta Guest   Posts: n/a Second-order ODEs may be integrated by either recasting them as first-order ODEs or simply using explicit Runge-Kutta Nystrom (RKN) methods. RKN methods are designed, from the start, for second-order ODEs. Here are some references: http://scholar.google.com/scholar?q=explicit+runge-Kutta+nystrom&ie=UTF-8&oe=UTF-8&hl=en (Sorry about the link - you'll have to manually enter it)

 January 27, 2005, 14:26 Re: Any questions about Runge-Kutta methods #7 Runge_Kutta Guest   Posts: n/a Yves, I have not used DG methods but where are you using the limiter; in the spatial or temporal discretization? From what I am aware of, many folks using DG methods like to use the 3-stage, 3rd-order ERK by Shu and Osher (JCP 1988). The currently trendy name for methods of similar design is SSP (Strongly Stability Preserving) methods. They do not have a limiter by they do possess a very specific and unusual stability property. Associated with this stability are the concepts of contractivity and monotonicity. Is this what you are asking about?? I can go about this but I'm worried that I'm not addresssing your question.

 January 27, 2005, 14:28 Re: Any questions about Runge-Kutta methods #8 Peter Attar Guest   Posts: n/a Thanks for the link..I was only attempting to explain why I thought Newmark and the such were used in structural dynamics..not that there is any justification behind it.

 January 27, 2005, 17:41 Re: Any questions about Runge-Kutta methods #9 noName Guest   Posts: n/a Can anyone comment on the conservative properties of an IRK or an IRK-Fehlberg, with local extrapolation and step size control? Do such schemes conserve momenta or energies or any other intergals of motion? Structural guys seem to use their own schemes because they conserve the right quantity and are dissipative for higher frequencies. See this paper: http://www2.mae.okstate.edu/Faculty/roy/hht4.pdf In addition to the reasons I mentioned, could someone tell me why structural people prefer Newmark type schemes instead of commonly used numerical methods that may be more efficient, of a higher order, better for stiff problems, etc?

 January 27, 2005, 18:26 Re: Any questions about Runge-Kutta methods #10 Runge_Kutta Guest   Posts: n/a First, the abbreviation IRK means Implicit Runge-Kutta. Fehlberg worked principally on explicit Runge-Kutta methods (ERK) and a form of mutiderivative Runge-Kutta methods. I don't recall ever seeing him deal with straightforward IRKs. Anyway, when one has a linear problem, one may speak about dispersion and dissipation orders of accuracy. With these methods, you design in the formal order of accuracy and the method will, hopefully, attain this order on any nonlinear problem. In addition, you design in higher-order accuracy for linear problems. These methods are sometimes referred to as reduced phase-error methods. You'll see these methods in acoustics. One can also do this with IRKs. These concepts are not exactly ones of conservation. Strictly speaking, local extrapolation (what Fehlberg did not do) and a good step size controller simply keep the local temporal error at some user specified value. The hope is that this will simultaneously also control the global integration error. Hence, your conservation is done to some order of accuracy. By the way, Tony Jameson's famous ERK is 4th-order on linear problems but only 2nd-order on nonlinear ones. Numerical methods can be strange. Applications people sometimes need a method but don't understand what relevant mathematicians have written. Their recourse has often been to simply try to invent methods by themselves. These methods are often unique to various small communities. The theory of phase-error in Runge-Kutta methods has been, in large part, described by Peter J. van der Houwen. He's got a bunch of stuff in SIAM J. Numer. Math. Otherwise, look for people who cite his work and offer useful methods. http://scholar.google.com/scholar?q=houwen+runge+kutta+phase&ie=UTF-8&oe=UTF-8&hl=en You should NOT assume that Newmark methods are the optimal methods for these type problems. I don't mean to sound like an arrogant SOB but there's definitely room for improvement here.

 January 28, 2005, 06:11 Re: Any questions about Runge-Kutta methods #11 Yves Guest   Posts: n/a I was indeed refering to, as it is called in the DG community, the TVB(Total Variation Bounded)-RK DG of Cockburn and Shu. See "Mathematics of Computation" and JCP 1989 - 1990. They use it for non-linear hyperbolic equations. Is such TVB/TVD property required for those equations? And what about linear equations as the LEE with constant mean flow? Do the requirements change for LEE with spatially varying mean flow? The equations are linear in the variables but the coefficients are not constant.

 January 29, 2005, 12:39 Re: Any questions about Runge-Kutta methods #13 Learner. Guest   Posts: n/a Hi, I would like to know why Third order Runge Kutta method is TVD? What does this mean in 'terms' of the terms used in the Runge Kutta?

 February 3, 2005, 20:21 Re: Any questions about Runge-Kutta methods #15 John Water Guest   Posts: n/a O.K. Do you have any reference about time error control? Just at a basic level.

 February 3, 2005, 23:40 Re: Any questions about Runge-Kutta methods #16 Runge_Kutta Guest   Posts: n/a I'm afraid the references I might give would bewilder most readers. So, I'll try to say it here. A Runge-Kutta method samples the space between two steps by creating intermediate solutions between the current step and future step. Let's say we want to integrate dU/dt = F(U) for one step begining at step n. Our first intermediate U-vector occurs at tn + c1*dt and is given by U^{1} = U^{(n)} + dt*[ a11*F^{1} + a12*F^{1} + a13*F^{3} + ...] Then the next intermediate U-vector, U^{2} occuring at tn + c2*dt, is U^{2} = U^{(n)} + dt*[ a21*F^{1} + a22*F^{1} + a23*F^{3} + ...] One does this s (s= number of stages) times, or U^{i} = U^{(n)} + sum_{j=1}^{s}[ dt*aij*F^{j} ] Now that you have all of the intermediate stages, you take a linear combination of intermediate function values to get U^{(n+1)}. We use b's for the weights used to get the step U-vector but a's to get the stage U-vector. U^{(n+1)} = U^{(n)} + sum_{i=1}^{s}[ dt*bi*F^{i} ] OK, once we've finished the step, we create a second solution that I'll call U-hat. The main solution is usually order p but the U-hat (the embedded method) is order (p-1). Hence we might have U^{(n+1)} = U^{(n)} + sum_{i=1}^{s}[ dt*bi*F^{i} ] Uh^{(n+1)} = U^{(n)} + sum_{i=1}^{s}[ dt*bhi*F^{i} ] where U^{(n+1)} might be 4th-order and Uh^{(n+1)} is 3rd-order. Create an error estimate by computing [U^{(n+1)} - Uh^{(n+1)}]/ [ U^{(n+1)} ] = delta^(n+1) Actually, one needs to design this error estimate to avoid dividing by zero. OK, now we have an error estimate. The step size is now selected using a controller and a desired error which I'll call eps. One selects the new step size to get us to step n+1 in the most general case as dt^(n+1) = safe_fac*dt^(n)* *( eps/delta^(n+1) )^alpha *( delta^(n)/eps )^beta *( eps/delta^(n-1) )^gamma *( delta^(n)/delta^(n-1) )^a *( delta^(n-1)/delta^(n-2) )^b If you are using an explicit Runge-Kutta method, Use a PI- or PID- controller (a=b=0). Start with alpha = 0.7/p_hat beta = 0.4/p_hat gamma = 0 or alpha = 0.6/p_hat beta = 0.2/p_hat gamma = 0 where p_hat is the order of the embedded method (3 in the case I have used). The only published PID-controller I've seen had alpha = 0.49/p_hat beta = 0.34/p_hat gamma = 0.10/p_hat Let me know if you want to use implicit RK methods because they use very different controller coefficients. Take a look at this paper http://ntrs.nasa.gov/archive/nasa/ca...1999099362.pdf The man who has done the most at error control for RK methods is Gustaf Soderlind. Read his (and his students) papers. http://www.maths.lth.se/na/staff/gustaf/ http://scholar.google.com/scholar?q=runge-Kutta+soderlind&ie=UTF-8&oe=UTF-8&hl=en Also, http://www.unige.ch/~hairer/books.html So, here's one method that I gave before: a(2,1) = 0.39175222657188905833d0 a(3,1) = 0.21766909626116921036d0 a(3,2) = 0.36841059305037202075d0 a(4,1) = 0.08269208665781075441d0 a(4,2) = 0.13995850219189573938d0 a(4,3) = 0.25189177427169263984d0 a(5,1) = 0.06796628363711496324d0 a(5,2) = 0.11503469850463199467d0 a(5,3) = 0.20703489859738471851d0 a(5,4) = 0.54497475022851992204d0 b(1) = 0.14681187608478644956d0 b(2) = 0.24848290944497614757d0 b(3) = 0.10425883033198029567d0 b(4) = 0.27443890090134945681d0 b(5) = 0.22600748323690765039d0 bh(1) = 3298630537163.d0/18795049059328.d0 bh(2) = 1771072646821.d0/14103719399933.d0 bh(3) = 25277699738943.d0/87361585864100.d0 bh(4) = 1305111367099.d0/5907422403405.d0 bh(5) = 3614287973383.d0/19159025146192.d0 I hope that wasn't too much ...

 February 11, 2005, 12:37 Re: Any questions about Runge-Kutta methods #17 shabnam Guest   Posts: n/a Hi i wrote a 4th order Runge-Kutta code in VBA Excel Macro and it is not fit for my experimental data at long residence times, so i would like to make a test for it and see the calculation parameters before using the main function but i don't know how can i make it, i will be very glad if some one can help me. Thank you very much in advance

 February 11, 2005, 13:41 Re: Any questions about Runge-Kutta methods #18 Runge_Kutta Guest   Posts: n/a I don't understand the question. You said the explicit? Runge-Kutta (ERK) method is not fit for your data at long times. What does that mean? Is the ERK failing? If so, what happens? What is it in the method that you would like to see fixed?

 February 15, 2005, 18:53 Re: Any questions about Runge-Kutta methods #19 John Water Guest   Posts: n/a Thank you very much! The references are very good. Do you know if some similar analysis has been carried out with Adam-Bashforth schemes?

 February 15, 2005, 19:55 Re: Any questions about Runge-Kutta methods #20 Runge_Kutta Guest   Posts: n/a Temporal error estimation and the controllers that adjust the step size are much more difficult for multistep methods than multistage methods. The best reference for this is a thesis from Sweden. Download the thesis by Anders Sjo here http://www.maths.lth.se/na/thesis.html I think you will conclude that doing error control with an explicit Runge-Kutta method is much easier than an explicit multistep method. The essential problem with error control with multistep methods is their history - they retain information from many old, and sometimes unequally spaced, steps.

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