
[Sponsors] 
August 18, 2008, 04:01 
CFLFree numerical methods

#1 
Guest
Posts: n/a

I want to use CFLFree 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: CFLFree numerical methods

#2 
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: CFLFree numerical methods

#3 
Guest
Posts: n/a

any good start then ?


August 18, 2008, 12:09 
Re: CFLFree numerical methods

#4 
Guest
Posts: n/a

Suppose you have an advectiondiffusion 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: CFLFree numerical methods

#5 
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: CFLFree numerical methods

#6 
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 NavierStokes, projection methods are appropriate for enforcing incompressibility. Good luck. 

August 19, 2008, 08:53 
Re: CFLFree numerical methods

#7 
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: CFLFree numerical methods

#8 
Guest
Posts: n/a

what is CIP ?


August 20, 2008, 06:03 
Re: CFLFree numerical methods

#9 
Guest
Posts: n/a

check Yabe or Xiao publication in JCP


Thread Tools  
Display Modes  


Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Numerical schemes for free surface flows (VOF)  botp  OpenFOAM  2  March 11, 2011 16:27 
numerical jacobian, matrix free methods  Nikhil  Main CFD Forum  1  December 5, 2008 04:51 
How can I find a free Fortran77 numerical library?  Alberto  Main CFD Forum  3  January 21, 2008 01:56 
1D numerical methods compared  Andrei  Main CFD Forum  0  October 14, 2002 03:22 
help:spectral methods & divergence free functionsn  D. Puigjaner  Main CFD Forum  1  August 28, 2000 10:06 