CFD Online Logo CFD Online URL
www.cfd-online.com
[Sponsors]
Home > Forums > Main CFD Forum

Questions for Trapezoidal method

Register Blogs Members List Search Today's Posts Mark Forums Read

Reply
 
LinkBack Thread Tools Display Modes
Old   February 15, 2008, 12:53
Default Questions for Trapezoidal method
  #1
jinwon park
Guest
 
Posts: n/a
I need to solve following ODE

dU/dt=S(U) in which S(U) is geometric source vector in symmetric Euler equations.

Due to instabilities resulting from stiffness, I am considering the use of an implicit time differential scheme. Among them, I first think about the Trapezoidal rule as

U^n+1=U^n+(S(U^n)+S(U^n+1))/dt.

The problem is that I do not know how to construct S(U^n+1). Could you anyone give me idea or direction to find books?

Thanks.
  Reply With Quote

Old   February 15, 2008, 15:30
Default Re: Questions for Trapezoidal method
  #2
Harish
Guest
 
Posts: n/a
You can expand S(U^n+1) using taylor series. Look up in any CFD book for implicit time marching.
  Reply With Quote

Old   February 15, 2008, 15:47
Default Re: Questions for Trapezoidal method
  #3
jinwon park
Guest
 
Posts: n/a
Do you mean that S(U^n+1)=S(U^n)+dS/dU*(U^n+1-U^n)? The problem in my implementations is that S consists of geometric term and any source term to impose sponge layer effects at boundaries. Thus, it was not easy to expand dS/dU for complex relations. Anyway thanks for your reply.
  Reply With Quote

Old   February 15, 2008, 16:42
Default Re: Questions for Trapezoidal method
  #4
rt
Guest
 
Posts: n/a
to solve u' = F(u,t), where functionality of F is not explicit in relation with u and t

there areseveral methods that only need time step size and a function that evaluate F at specific "u" and "t", e.g. Rungee Kuta methods (http://en.wikipedia.org/wiki/Runge-Kutta_methods)

if you can not evaluate F at mid way of time step (RK needs), you could take its as picewise constant or use one-sided interpolation from avaialble data.

  Reply With Quote

Old   February 15, 2008, 18:53
Default Re: Questions for Trapezoidal method
  #5
jinwon park
Guest
 
Posts: n/a
Did you mention it for implicit schemes? I needed to implement an implicit scheme to continue the computation beyond a certain instant. I am currently suffering from stability problems when the compressible Euler code with a geometric source applied to simulate a wide range of fluid flows.

Thanks in advance!
  Reply With Quote

Old   February 15, 2008, 20:36
Default Re: Questions for Trapezoidal method to rt
  #6
jinwon park
Guest
 
Posts: n/a
As you mentioned, I obtained interpolated values in implicit method as

U^n+1=U^n+1/2*dt*(S(u^n)+S(u*^n+1))

where u*^n+1 is interpolated from previous solutions. The procedure to interpolate is based on

U*^n+1=U^n+(U^n-U^n-1). This is my idea to obtain values. Did you intend to let me know this way? Or is there any other way? I really appreciate your great advice
  Reply With Quote

Old   February 15, 2008, 22:46
Default Re: Questions for Trapezoidal method to rt
  #7
rt
Guest
 
Posts: n/a
suggested methods are explicit, but with good stability bound, note that the R-K method are like predictor-corrector methods and they try to reconstruct function with higher order and lower needed data, i should say that your extrapolation is also somhow explicit and it is like Adams-Bashforth method which is bit unstable than 4th-R-K, so try R-K methods first, they may resolve. They are extremly simple to implement.

i suggest extrapolation to evaluate f(u,t) at midway. i.e.. at (u+du, t+dt), where u+du<n^{n+1} t+dt<t^{n+1}

what is structure of "F", i do not understand what you mean from "geometric"?

do you have IMSL library, if, it has several prepared routine inter/extrapolation and ODE solvers.
  Reply With Quote

Old   February 16, 2008, 01:09
Default Re: Questions for Trapezoidal method
  #8
Giovanni Lanzillo
Guest
 
Posts: n/a
If you know S(u) and if you know U initial value why you dont'use R-K?
  Reply With Quote

Old   February 16, 2008, 01:19
Default Re: Questions for Trapezoidal method
  #9
jinwon park
Guest
 
Posts: n/a
I used RK scheme but it caused instabilities when the simulation went long. At early time(within sub-microsecond), it worked well. But at relatively late time(around sub-millisecond), the computation failed to continue after a certain instant. I guessed that this instability results from stiffness of ODE which I was solving. The ODE is a part of compressible Euler equation with a geometric source implementing symmetry spherically or cylindrically. Unfortunately, it leaded to the sudden abortion of computation when I used RK scheme to numerically solve that ODE. It is

dU/dt=S(u) where U=[rho, rho*u, rho*E]^T and S(u)=1/r[rho*u,rho*u^2m rho*(E+p)]^T. This is a simplified version of compressible Euler equation implementing symmetric flow motion. I did not know about this difficulty before. However, when I attempted to see late-time flow motion, I had to run the compressible code somewhat long and I met with sudden abortion of computation due to several reasons. I analyzed that the stiffness of ODE is a main source of this instability. To overcome this issue, I tried to implement an implicit time integration which is free from CFL restriction.

If you have any other advices, could you give me that? I am looking forward to having change to talk all together.

Thanks.

  Reply With Quote

Old   February 16, 2008, 08:16
Default Re: Questions for Trapezoidal method
  #10
rt
Guest
 
Posts: n/a
if you need a really implicit solver, you could perform an outer iteration within each time step and go uot when your predicted u^n+1 converge.

there are several implicit ode solver but cost is high, e.g. see Adamsâ€"Moulton methods

http://en.wikipedia.org/wiki/Linear_multistep_method

if you look at a classic numerical analysis book found formal information needed to solve ode

e.g. this one is good: Numerical Analysis, R. Burden
  Reply With Quote

Old   February 18, 2008, 06:52
Default Re: Questions for Trapezoidal method
  #11
rt
Guest
 
Posts: n/a
you may find this paper interesting:

http://www.global-sci.com/freedownload/v3_479.pdf

  Reply With Quote

Reply

Thread Tools
Display Modes

Posting Rules
You may not post new threads
You may not post replies
You may not post attachments
You may not edit your posts

BB code is On
Smilies are On
[IMG] code is On
HTML code is Off
Trackbacks are On
Pingbacks are On
Refbacks are On


Similar Threads
Thread Thread Starter Forum Replies Last Post
Some questions about CE/SE method Chris Main CFD Forum 3 January 14, 2013 11:16
discretizer - gmshToFoam Andyjoe Open Source Meshers: Gmsh, Netgen, CGNS, ... 13 March 14, 2012 05:35
Fluent 6.3.26 vs 12.1 and partition method Anorky FLUENT 0 April 27, 2010 10:55
Code for most powerfull FDV Method D.S.Nasan Main CFD Forum 6 September 4, 2008 02:08
hess-smith method and fvm method yangqing FLUENT 0 March 20, 2002 19:25


All times are GMT -4. The time now is 07:52.