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joy August 18, 2008 04:01

CFL-Free numerical methods
 
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....


Jed August 18, 2008 08:35

Re: CFL-Free numerical methods
 
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.

joy August 18, 2008 09:15

Re: CFL-Free numerical methods
 
any good start then ?

Jed August 18, 2008 12:09

Re: CFL-Free numerical methods
 
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.

joy August 19, 2008 04:39

Re: CFL-Free numerical methods
 
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?


Jed August 19, 2008 05:31

Re: CFL-Free numerical methods
 
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.

vadim August 19, 2008 08:53

Re: CFL-Free numerical methods
 
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


student August 20, 2008 05:40

Re: CFL-Free numerical methods
 
what is CIP ?

vadim August 20, 2008 06:03

Re: CFL-Free numerical methods
 
check Yabe or Xiao publication in JCP


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