Does anyone have any questions about IMPLICIT 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?

Hi Runge-Kutta,

Do you have any references/examples of the use of large scale CFD problems (esp. of LES/DNS) using 4th order implicit RK methods?

1) Are they stable for problems involving acoustics and heat release coupling (combustor dynamics)? 2) Do they present significant advantage in wall-clock time for large Acoustics-LES problems? 3) How do they affect convergence? 4) Are there any single step 4th order accurate RK methods that can be applied to large problems?

Do you have any publications that we can look into?

Sachin

Hi Sachin,

I am only aware of a few folks that use 4th-order integrators in CFD where there is some degree of implicitness and the methods are Runge-Kutta. Most of it has it's origin at NASA Langley. If you want a fully implicit formulation then here are some papers for you

http://fun3d.larc.nasa.gov/papers/carpenter.pdf ftp://techreports.larc.nasa.gov/pub/...-2001-2612.pdf http://www.hsa.lr.tudelft.nl/publica.../2002.2.01.pdf

The ESDIRK methods are very accurate in this context but get into a bit of trouble during the iterations. This is because they allow such large step sizes that the multigrid or inexact Newton iteration often begins with a relatively poor starting guess. There are things called stage value predictors that are specifically designed to enhance the quality of this starting guess. However, there has yet to be a serious attempt to optimize stage-value predictors with any ESDIRK. Also, see this for other methods:

http://www.math.ntnu.no/preprint/num...004/N1-2004.ps

A second context for implicit Runge-Kutta (IRK) methods in CFD is when you decide to integrate, say, diffusion implicitly but convection explicitly. Here's a use for fluid-structure interaction.

http://www.sdu.dk/Nat/MCI/rm/wcna200...s/cppp-i27.pdf

If you have reactive stiffness, you could also use them.

By the way, all of what I say here is specifically targeted to the compressible equations and not to the DAEs found in the incompressible and the low-Mach number equations.

There is so much to say about stability, I don't know what to say within this terse format. You can design in good stability characteristics to implicit Runge-Kutta methods if you know what you are doing. If you use one, make sure it was designed well. You're looking for methods that are L-stable. A-stability is not good enough. Algebraic stability doesn't buy you much. Stage-order is the key. On stiff problems, the IRK method will exhibit a convergence rate somewhere between the classical order and the stage-order plus 1. The ESDIRKs above have formal order 4 but stage-order 2 so that they may crumble a bit under stress. There is still room for improvement in some of these methods. By the way, the common approach to situations like this are the BDF family of methods. BDF2 is fine. BDF3 is what we call L(alpha)-stable. In CFD, you will generally find that convection term will destabilize BDF3 because of the small instability lobes of BDF3 near the imaginary axis of its stability domain.

1) An L-stable ESDIRK can work with any stiffness that you give it. Converging the iterations at very large step size may be difficult but that is mostly a function of the quality of the starting guess. Acoustics will strain the method in the imaginary axis of its stability domain while combustion generally strains it on the negative real axis.

2) One thing people need to keep in mind here is that integrations must be done at some chosen (and reasonable) error tolerance. That is THE essential point. Once you decide what error tolerance you need, you then set about the task of finding the method to deliver this tolerance as cheaply and reliably as possible. As your accuracy requirement becomes more stringent, higher-order methods become more appealing. It is not at all uncommon to see users pushing the time step so large that the integrator no longer has it's design convergence rate.

3) Convergence is influenced by a million things. A good L-stable ESDIRK should work as well or better than most other fully implicit CFD methods.

4) Most of the ESDIRKs in the references that I have included are 4th-order and all Runge-Kutta methods are single step methods.

By the way, there are many good classes of IRK methods but when you apply them to millions of equations then the choices narrow. You could always try the two-stage Radau IIA, or Burrage and Butcher's SIRK methods. I do not recommend DIRKs or SDIRKs. They have only stage-order 1 - and that is not good. Other fully implicit RK methods like Gauss an Lobatto should be bypassed. I think all of the mono-implicit RK methods would be way too expensive to implement and are only of interest if you use a stage-order 3 version.

Last thought: If you're going to use an implicit method then use an error estimator to avoid kidding yourself about the accuracy of your solution. There are some pretty good methods out there but the quality of your starting guess for the iterations and the iterative solver can totally ruin the party if they suck.

Hi Runge Kutta,

Thanks for your detailed response. I am very impressed by your work. If you are still interested in looking for a job, please send me your resume at my email address. For reasons of avoiding mass emailing and calling, I am not disclosing my work details in an open forum. I can discuss about the job details outside the forum.

Sachin