# CFL-Free numerical methods

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

 August 18, 2008, 04:01 CFL-Free numerical methods #1 joy Guest   Posts: n/a I want to use CFL-Free numerical methods. I did google it and come across interesting staff but still I could not understand it. Can you explain how can we use it, is it for all type of PDEs. what is its relation to centeral methods....

 August 18, 2008, 08:35 Re: CFL-Free numerical methods #2 Jed Guest   Posts: n/a Central versus upwinded is a stability issue separate from CFL. To avoid CFL, you need to use an implicit method in which case you can choose arbitrarily long time steps. Of course, if the time steps are much longer than CFL, you may not resolve temporal dynamics of the system.

 August 18, 2008, 09:15 Re: CFL-Free numerical methods #3 joy Guest   Posts: n/a any good start then ?

 August 18, 2008, 12:09 Re: CFL-Free numerical methods #4 Jed Guest   Posts: n/a Suppose you have an advection-diffusion equation du/dt + a grad(u) - e div(grad(u)) = 0 in Omega u = 0 on Gamma where a is the velocity field, e is viscosity. More generally this is written du/dt + F(u) = 0 Discretize in time using any unconditionally stable ODE method. With backward Euler we get G(u^{n+1}) = u^{n+1} - u^n + dt F(u^{n+1}) = 0 This is a linear (or nonlinear) system for u^{n+1}. In the general case, it will require a Newton (or Picard) iteration to solve. Let J(u^n) = (dG/du)(u^n) be the Jacobian (discretized Frechet derivative) and solve solve linear systems J(u^n + v_k) v_{k+1} = -G(u^n + v_k) starting from v_0=0 until convergence. Then u^{n+1} = u^n + v_k and move on to the next time step. Note that you only need one Newton step when the problem is linear, hence u^{n+1} = u^n + v_1.

 August 19, 2008, 04:39 Re: CFL-Free numerical methods #5 joy Guest   Posts: n/a Many thanks for your response and time, very much appreciated. Am doing it now and I will see if I can get done with fortran. Just need to know then about the nature of the PDEs i.e. are we really need to confirm hyperbolicity of the equations and the Jacobian or just consider any system and solve it e.g. Euler equations. What do you think then?

 August 19, 2008, 05:31 Re: CFL-Free numerical methods #6 Jed Guest   Posts: n/a In advection dominated systems, you will need some form of upwinding to stabilize the advection term. You need to know what boundary conditions are necessary and sufficient (this is related to hyperbolicity which is a concept which is only really meaningful in the linear case). If you are solving constrained equations (incompressible) using an iterative method, it will require extra work due to it being a saddle point problem. For incompressible Euler equations and modest to high Reynolds number Navier-Stokes, projection methods are appropriate for enforcing incompressibility. Good luck.

 August 19, 2008, 08:53 Re: CFL-Free numerical methods #7 vadim Guest   Posts: n/a CFL is related to convection/advection, so CFL free method is which has not time step limitation due to stability. CFL free methods: implicit treatment of convection can be stable (but handling nonlinearity is bit difficult and make method inefficient) beter methods are semi lagrangian methods, e.g. CIP

 August 20, 2008, 05:40 Re: CFL-Free numerical methods #8 student Guest   Posts: n/a what is CIP ?

 August 20, 2008, 06:03 Re: CFL-Free numerical methods #9 vadim Guest   Posts: n/a check Yabe or Xiao publication in JCP

 Thread Tools Display Modes Linear Mode

 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 OffTrackbacks are On Pingbacks are On Refbacks are On Forum Rules

 Similar Threads Thread Thread Starter Forum Replies Last Post botp OpenFOAM 2 March 11, 2011 16:27 Nikhil Main CFD Forum 1 December 5, 2008 04:51 Alberto Main CFD Forum 3 January 21, 2008 01:56 Andrei Main CFD Forum 0 October 14, 2002 03:22 D. Puigjaner Main CFD Forum 1 August 28, 2000 10:06

All times are GMT -4. The time now is 05:10.