Hi,

I am solving the uncompressible time dependent NS-Equations using MAC algorithm and it is fine with explicit Euler's 1st order time marching scheme. I am trying to implement low storage explicit Runge-Kutta 3rd order (RK-3, 5-stages) scheme. In Euler scheme, I solve Pressure Poission Equation (PPE) to get P(t) and then advance to V(t+dt) using this P(t) as described in the classical MAC algorithm. However, while using RK-3, I need to solve the PPE 5-times and at end of the 5-th stage, I get V(t+dt) passing though many intermediate stages between V(t) and V(t+dt).

The things I am not able to understand is:

1. Pressure field at what stage (out of the 5 stages) should be taken as true P(t), and

2. Will the PPE be different in case of RK-3 than what was in case of Euler's time marching scheme?

I think people who have used RK methods to solve NS equations would be having a good idea about this issue. Please suggest me a little about it.

Thank you very much in anticipation.

-Chandra

Application of a fractional-step method to incompressible Navier-Stokes equations J Kim, P Moin - Journal of Computational Physics, 1985

This paper discusses the issue in detail.The fractional step algorithm is based on the MAC scheme.

Thank you very much Harish. I will look into the paper u mentioned.

Here are a couple of comments:

1) Runge-Kutta methods are multistage methods.

2) (Linear) Multistep methods are the class of methods which contain Adams-Bashforth, Adams-Moulton, BDF methods, etcetera. When the past information is reformulated in terms of a Taylor series rather than old values of the RHS evaluations , F(U), or U, these are called Nordsieck methods.

3) Multistep Runge-Kutta methods combine 1) and 2). They are a subclass of "General linear methods." Look for papers by Kevin Burrage on Google Scholar.

4) The semi-discretized incompressible NSE are not ODEs but differential algebraic equations (DAEs). In their unmolested form, they are index-2 DAEs. When the constraint equation is differentiated, this is the index-1 form.

5) Explicit Runge-Kutta methods can have a hard time with DAEs. Go look at the book by Gresho and Sani. Consider "half-explicit" Runge-Kutta methods like those from Martin Arnold and Ander Murua.

6) For the incompressible NSEs, pressure is the "algebraic variable," i.e. z.

7) If you really want to understand how to integrate DAEs, look at Hairer and Wanner (1996).

Thank you very much Runge_Kutta for the useful information. As far as I understand, NS equations are not ODEs and that is why in principle Runge-Kutta method can not be applied. However, after discretizing RHS of the momentum equations, it converts into ODE, say du/dt = f(u,v,w,P) and that is why we can apply the Runge-Kutta approach to NS equations as well, explicitly or implicitly. In fact I could manage to device an algorithm for PPE formulation using RK4 after understanding the concepts hidden behind the MAC algorithm and the Runge-Kutta algorithm. However, my concerns are related to implementation of Williamson type low storage RK schemes but unfortunately I could not find a text which focuses on basics of the concept hidden behind low-storage RK-schemes. General RK-scheme is well described on Wikipedia but not the Low-storage one. If you have any link/reference for this, it would really be very helpful for me.

I could use the low storage RK4 for Energy Equation solution only after following the algorithm of the scheme, but I am having hard time in writing an algorithm for Low storage RK4 for momentum equations using MAC following PPE formulation.

Thank you very much,

regards,

Chandra

Chandra,

The NSE for compressible fluids becomes an ODE system after semi-discretization. The incompressible and low-Mach number formulations become DAEs.

Low-storage explicit Runge-Kutta methods were conceived at a time when computers had small memories. They were particularly beneficial in extremely lean DNS codes where the memory used per grid point was small. Computer memories are much bigger now and most codes devote a relatively small amount of memory to the integrator. Besides, low-order ERKs don't use much memory anyway. However, if you can't live without them, the two major approaches were introduced here:

1) Low-storage Runge-Kutta schemes WILLIAMSON, J H Journal of Computational Physics. Vol. 35, pp. 48-56. 15 Mar. 1980

http://scholar.google.com/scholar?q=williamson+%22runge-Kutta%22&hl=en&lr=&btnG=Search

2) Peter J. van der Houwen, Construction of integration formulas for initial value problems, page 42, (1977) [Book]

People who solve DAEs should really read up on them. Again, try Gresho and Sani. If you are solving isothermal incompressible flows, life isn't so bad. If not, you really need to understand DAEs and you absolutely must get your constraint right.

Thank you very much. This info is going to be quite useful. I was under impression that incompressible NS Equations also converts into an ODE after discretization of spatial derivatives.

I also have to deal with the DNS stuff and that is why I was paying attention to low storage RK schemes. But I do not have much idea if computer memories nowdays are sufficient so that one can do DNS simulations even if general RK schemes are used, and in this regard I want to have your suggestion. Until how many units of memory can be allocated per variable (if low storage RK is used, 2 units per variable) ??

If I can use general RK, I absolutely do not need to dig for the low storage RK.

I am dealing with the energy equation as well and so of course temperature of the liquid would be varying. Physical properties of the flowing liquid is considered independent of temperature for simplicity.

Thank you again for your useful information,

Chandra

For the low storage stuff, go to Google scholar and type in "low storage" and "Runge-Kutta."

http://scholar.google.com/scholar?q=low+storage%22+%22runge-kutta%22&hl=en&lr=&btnG=Search

Now you can research the matter.

I have never thought about constraint manifolds for the non-isothermal NSE in liquids. This may actually be easier because the medium is almost incompressible. The basic idea, though, is that there is a manifold that expresses the elimination of the fast acoustic time scale. In gases that have species of differing molecular weights or non-isothermal flows, this manifold is more complicated than Div(U)=0. If you integrate your equations on the wrong manifold, the acoustic energy is not suppressed properly. Explicit Runge-Kutta methods have trouble with DAEs because the order of accuracy of the stage approximations are low. This is referred to as low "stage-order." Some implicit RK methods have very high order and stage order but they are every expensive to implement.

For incompressible flows where diffusive or reactive time scales (numerical) are on the order of the convective time scale are non-stiff. For these the modified Half-Explicit RK methods of Martin Arnold and Ander Murua are potentially useful. The first hit is a down-loadable .ps file.

http://scholar.google.com/scholar?nu...E+arnold+murua+%22runge-kutta%22&btnG=Search

When they talk about partitioned methods, they mean that the differential equations are updated with one method but the constraint is enforced with a different (implicit) method. Your algebraic variable, z, is the pressure. The differential variable, u, is the vector of variables like velocity, temperature, and whatever else. Your constraint equation, g(u) or gamma(u), is something resembling Div(U)=0 but maybe something more complicated. DAEs are not for the faint of heart!!

OK, good luck. I can't spend any more time on this.

Thank you for your time Runge_Kutta and the useful information you supplied. The information seems to be enough to start my own digging and get familiar with deep hidden things.

Thank you very much,

Chandra