CFD Online URL
[Sponsors]
Home > Forums > Main CFD Forum

Main advantage of using Runge Kutta of higher order?

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

Reply
 
LinkBack Thread Tools Display Modes
Old   May 14, 2013, 05:41
Default Main advantage of using Runge Kutta of higher order?
  #1
New Member
 
Jakub
Join Date: May 2013
Location: Czech Republic
Posts: 16
Rep Power: 3
jakubstary is on a distinguished road
What is main advantage of using 2nd and 4th order Runge Kutta methods for time discretization? Is it stability for larger CFL condition or Runge Kutta is more accurate for same CFL condition?

I compute Shallow water equations and I use Euler method, RK2 and RK4 for time discretization. If I have CFL = 0.9 then I get Euler method as the most accurate, but if I set CFL = 3.5 then I get RK4 is more accurate than Euler method with CFL=0.9. Euler method is obviously unstable for CFL = 3.5.

Are my results correct?

Thank you
Jakub
jakubstary is offline   Reply With Quote

Old   May 15, 2013, 04:28
Question
  #2
New Member
 
Jakub
Join Date: May 2013
Location: Czech Republic
Posts: 16
Rep Power: 3
jakubstary is on a distinguished road
I have tested the Euler method and the RK2 method for increasing CFL and I get this result. Don't anyone know if it's correct result? If not, don't you have an idea where might be error in my calculations?

Thank you.
Attached Images
File Type: jpg swe_cfl_test.jpg (46.3 KB, 25 views)
jakubstary is offline   Reply With Quote

Old   May 17, 2013, 07:29
Default
  #3
Senior Member
 
Lefteris
Join Date: Oct 2011
Location: UK, Greece
Posts: 186
Rep Power: 5
Aeronautics El. K. is on a distinguished road
Euler method is first order accurate while RK4 is forth order accurate. Moreover, the explicit Euler method has quite strict stability criteria.
__________________
Lefteris

Aeronautics El. K. is offline   Reply With Quote

Old   May 17, 2013, 07:43
Default
  #4
New Member
 
Jakub
Join Date: May 2013
Location: Czech Republic
Posts: 16
Rep Power: 3
jakubstary is on a distinguished road
And is it correct, that Euler is more accurate for small CFL condition than RK2 and RK4? I thought RK2 is always more accurate than Euler, but in my program not, see graph.

Thank you very much.
Jakub
jakubstary is offline   Reply With Quote

Old   May 17, 2013, 07:59
Default
  #5
Senior Member
 
Lefteris
Join Date: Oct 2011
Location: UK, Greece
Posts: 186
Rep Power: 5
Aeronautics El. K. is on a distinguished road
e is the temporal discretisation error?
__________________
Lefteris

Aeronautics El. K. is offline   Reply With Quote

Old   May 17, 2013, 08:09
Default
  #6
New Member
 
Jakub
Join Date: May 2013
Location: Czech Republic
Posts: 16
Rep Power: 3
jakubstary is on a distinguished road
e is sum L1 error between numerical solution and exact solution of Riemann problem.
jakubstary is offline   Reply With Quote

Old   May 17, 2013, 08:47
Default
  #7
Senior Member
 
Lefteris
Join Date: Oct 2011
Location: UK, Greece
Posts: 186
Rep Power: 5
Aeronautics El. K. is on a distinguished road
what problem are you solving? NS or Euler Equations? How is the spatial discretisation done?
__________________
Lefteris

Aeronautics El. K. is offline   Reply With Quote

Old   May 17, 2013, 09:03
Default
  #8
New Member
 
Jakub
Join Date: May 2013
Location: Czech Republic
Posts: 16
Rep Power: 3
jakubstary is on a distinguished road
I'm solving Euler equations. For case which is on graph I used simple Lax-Friedrichs scheme for spatial discretisation.
jakubstary is offline   Reply With Quote

Old   May 17, 2013, 10:57
Default
  #9
New Member
 
Jakub
Join Date: May 2013
Location: Czech Republic
Posts: 16
Rep Power: 3
jakubstary is on a distinguished road
Please, Aeronautics El. K. don't you have any idea, why Runge Kutta behave strangely for my case?

Thank you.
jakubstary is offline   Reply With Quote

Old   May 17, 2013, 11:16
Default
  #10
Senior Member
 
Lefteris
Join Date: Oct 2011
Location: UK, Greece
Posts: 186
Rep Power: 5
Aeronautics El. K. is on a distinguished road
I can't make anything of it yet. I'm reading a little bit on it and I suggest you do the same
__________________
Lefteris

Aeronautics El. K. is offline   Reply With Quote

Old   May 17, 2013, 11:32
Default
  #11
New Member
 
Jakub
Join Date: May 2013
Location: Czech Republic
Posts: 16
Rep Power: 3
jakubstary is on a distinguished road
Aeronautics El. K.: Of course I do the same :-) I'm trying to solve it for about two weeks :-(. Thank you for your willingness to help.

If someone will have some idea or tip for book, where can I find it, please write it here. Many thanks.

Jakub
jakubstary is offline   Reply With Quote

Old   May 17, 2013, 19:32
Default
  #12
New Member
 
Jakub
Join Date: May 2013
Location: Czech Republic
Posts: 16
Rep Power: 3
jakubstary is on a distinguished road
I discovered an interesting thing. I found Masatsuka's code http://www.cfdbooks.com/cfdcodes/oned_euler_v1.f90, where he solve same problem as me. I implemented even Euler method to his code. I compared results obtained using RK2 and Euler method for time discretization and Euler is more accurate. So really advantage of using RK2 and RK4 instead of Euler is probably only possibility using larger timestep (or CFL)? I think that RK2 and RK4 may not be always more accurate.

I don't if this is correct conclusion of my problem :-). Books say something else, but numerical results not.

Jakub
jakubstary is offline   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
2nd order boundary conditions for 2nd order discretization? quarkz Main CFD Forum 30 December 26, 2011 08:12
1D Burgers euqation with 4th Runge Kutta dokeun Main CFD Forum 3 August 8, 2011 07:34
fractional step method with third order rugne kutta TVD HaKu Main CFD Forum 1 October 18, 2009 20:33
Higher order discretization on staggered grid Chandra Shekhar Main CFD Forum 9 January 27, 2005 17:31
Higher Order FV Schemes for unstructured meshes Apurva Shukla Main CFD Forum 4 December 15, 2000 10:17


All times are GMT -4. The time now is 18:52.