I am starting out my endeavors with higher than second order time integration of the Navier Stokes equations for a pressure-based approach (SIMPLE type schemes). I was wondering if somebody could give some references which discuss the advantages and disadvantages of the various techniques available for such a time integration.

One specific question regarding the Runge Kutta methods that I am trying to resolve these days is if anybody has attempted to create a single stage RK implicit formulation?

Are there single-stage/single-step formulations which can lead to higher than second order integration?

Sachin,

A few comments. Implicitly, you are speaking of solving systems of index-1 or index-2 DAEs.

At the broadest level, one has three different ways to go beyond Euler's method.

1) A multistage method samples the space between t_{n} and t_{n+1} with many stages to update the solution at t_{n+1}. These are more commonly known as Runge-Kutta methods.

2) A multistep method uses historical information from previous steps to assist in updating the solution at t_{n+1}. This is your Adams, BDF, etc. methods.

3) A multiderivative method uses not only the values from the right-hand-side of the equations but also uses their time derivatives. These also go by the name Taylor series methods.

OK, now one is free to combine these:

Turan methods = multistage, multiderivative methods

Obreshkov methods = multistep, multiderivative methods

General linear methods = multistep, multistage methods

Since the exact value of dF/dU is required for second-derivative Turan and Obreshkov methods, this is rather hopeless for huge systems of equations like the NSE.

dU/dt = F(U)

0 = g(U) or 0 = g(U,p)

One could also design multistage, multistep, multiderivative methods but these are INSANELY complicated. There are maybe 5 papers on the topic in the last 30 years [By Hairer and Wanner(1973) and later by Burrage].

There is a general feeling that the most useful broad class of methods for something like the NSE are General linear methods. These are still alarmingly complicated and studied by few people.

Getting back to your question, possibly prompted by AMR issues, if you insist on a single-stage/single-step method AND high-order then you must use a multiderivative formulation. Actually, you could allow the coefficients of the method to be a function of the RHS but that would be the beginning of a horrible nightmare.

Hi Runge_Kutta,

Thanks for your well informed response.

I have been reading on time integrations recently and the things that I have noted are as follows:

In lay man terms: 1) Implicit schemes are better for convergence, primarily because of the way they bound the solution. In terms of matrix algebra, implicit solution approach involves coupling the last time step to the current time step and provide a controlled growth of the solution in time.

2) You are right about the motivation of single-stage schemes to be AMR. It reduces a lot of computational effort if we can spend considerable time upfront and design a single stage implicit scheme. With the availability of advanced solvers which do not rely on diagonal dominance (even in parallel), it seems that this effort upfront might be worthwile in the long run, if we can come up with a stable and consistent single-stage higher order formulation

3) The idea of higher order space integration typically, in FV sense, involves reconstructing a higher order polynomial which can be solved on discete points (typically gauss points on the faces of a CV) and the numerical integration of the flux functions can be completed through Gauss quadrature or any other numerical integration approach. The higher order integration requires the evaluation of derivatives of the Taylor series, which are very ingenuisely done by Dr. Gooch in his Least Squares based higher order reconstruction (although motivated by Barth and Venkatkrishnan seperately in 1993). I was hoping somebody might have tried a similar approach for time integration.

4) In response to your advise above, you indicate a multiderivate alternative to be a nightmare. Is it because you feel it will give us stability and convergence and consistence issues (even if well thought and worked hard upon upfront) or is it because it will involve a lot of complex maths and we may never be able to formulate it. or is it because of the computational cost we might incur when trying to solve it? or is it something else. It will be very kind of you, if you could respond to this point.

Thanks

Hi Sachin,

1) Implicit methods are distinguished from explicit methods principally by what is possible with their stability functions. The stability function controls the propagation of global error. When the stability function of a Runge-Kutta method exceeds one in absolute value, there is a geometric growth in global error that is independent of the local error committed on a given step. Explicit methods have very limited forms of their stability functions. I'll leave it at that unless you want more.

2) Your menu includes any general linear method (GLM) but nothing else. Multiderivative methods are hopeless for general, fully-implicit, NSE computations. Think of three axes, similar to x, y, and z. Let the x-axis be the number of stages. Let the y-axis be the number of steps and let z be the number of derivatives. You would be wise to live in the x-y plane. If you want high-order, you may proceed out either axis or may move into the center of the plane. Anyway, if these methods were easy, everyone would be designing them and using them. They are not. GLMs are VERY complicated beasts. The only popular ones that see any real use are MEBDF methods. MEBDF methods live close to the y-axis in my analogy. Try changing the step size. Try designing a good error controller for them. Try designing a dense output method. Most of all, compute their stability function in the case of an arbitrary step size sequence. Which sequences maintain L-stability and which do not.

3) Higher-order implicit time integration methods and fluid dynamics do not have much history together.

4) For second-derivative Turan or Obreshkov methods, you need to rewrite dF(U)/dt as (dF/dU)*(dU/dt) = (dF/dU)*F. Are you willing to provide the exact jacobian for a RHS of several million equations? You may not approximate this term any more than you can F. This is hopeless for a fully implicit NSE solver. Let's not forget you must also derive this jacobian analytically!! Moving out on the z-axis (multiderivative) makes sense if a) you have a small system of equations, and b) the jacobian is readily formulated. There are tons of papers on second-derivative BDF methods (an Obreshkov method). You can make BDF methods L-stable to 4th-order rather than to just second order for (first-derivative) BDF methods. It is not a practical approach for NSE solvers.

http://scholar.google.com/scholar?hl...ultiderivative+multistep

Gratuitous remarks: It is a good bet that there are existing methods out there that are better than what you are currently using. I would recommend trying those rather than venturing out into terra incognito. Alternatively, you could fund research into better implicit methods destined for the NSEs. Better yet, find the people who dispense funding at NSF or the Office of Science at DOE and tell them what you need.

Hi Runge_Kutta,

Thanks a lot for your detailed and very helpful comments. I hope maybe I can answer some questions for you someday.

In reference to the point (1) that you mentioned, yes, could you please talk more about the stability functions when you get time and provide some initial reference which might help me get to speed.

Thanks

Hi Sachin,

Here are some more thoughts on stability. As you search for your favorite implicit method, consider linear stability, nonlinear stability, and internal stability. Most discussions of stability for implicit methods focus on linear stability. The best reason for this is that once you're L-stable, the other two measures are of much lower importance. By the way, multistep methods do not have "internal stability," it only arises if there is a multistage character to the method. Many people have focused on nonlinear stability by making methods "algebraically stable." If you can get this for free, take it. Otherwise, it is not clear where it has any genuine value with first-order ODEs. If you really want to learn about all of this, go to your library and get:

http://www.springeronline.com/sgw/cd...1137-0,00.html

http://www.springeronline.com/sgw/cd...9809-0,00.html

http://www.springeronline.com/sgw/cd...7688-0,00.html

http://www.amazon.com/exec/obidos/ASIN/0387518606/qid%3D1111391958/sr%3D11-1/ref%3Dsr%5F11%5F1/002-9287138-4896024

http://www.wiley.com/WileyCDA/WileyT...471967580.html

http://www.amazon.com/exec/obidos/tg...glance&s=books (1987 Edition)

http://www.amazon.com/exec/obidos/tg...glance&s=books

http://www.amazon.com/exec/obidos/tg...glance&s=books