What is the best way to calculate the CFL-based timestep in an unstructured grid? Thanks a lot!

ma

Dear ma,

The CFL based time step in an unstructured grid can be calculated as follows.

dt = CFL * sl/[abs(q)+a]

where dt is the time step

CFL is the Courant number

sl is some characteristic length, typically the square root of area in 2D or cube root of volume in 3D q is the total velocity( q=sqrt(u**2 + v**2) ) where u and v are the velocity components(2D). and a is the sonic speed.Note that [abs(q)+a] is the largest eigen value.

Note that dt is calculated for ecah cell in the domain.

It is possible to use two types of time stepping procedures.

1. Global time stepping.

The minimum of dt calculated for all cells is used for the stepping in the entire domain. As you can easily appreciate, the convergence would be slow in this case.

2. Local time stepping.

The dt calculated in each cell is used for stepping in the respective cells. A good acceleration convergence techinique esp. for viscous flows where you experience grid-induced stiffness. This is generally recommended for steady flow problems. In conjuction with implicit schemes, is a very fast convergence prompting tool. However, cannot be used for unsteady flows as such.

Hope this information helps.

Regards,

Ganesh

A slight variation on

... dt = CFL * Lref / max_eigenvalue

for a cell-centered finite volume method is

... dt = CFL * vol(i) / [sum_{k=1,nfaces(i)} lambdaA(k)]

where

i = cell index

k = face index

nfaces(i) = number of faces for cell i

lambda(k) = u(k)*Ax(k) + v(k)*Ay(k) + w(k)*Az(k) + a(k)*A = max eigenvalue based on (velocity dot area vector)

u,v,w = Cartesian velocity components

Ax,Ay,Az = Cartesian component of area vector normal to face k

A = sqrt(Ax**2 + Ay**2 + Az**2) = area of face k

a(k) = speed of sound based on local properties of face k

Thanks a lot for these details.

- ma