CFL condition for higher order schemes
 

Hi,

I read in an article that the CFL condition for higher order schemes can be very low, as the resolution is given by (h/p^d), where h is the grid spacing, p is the polynomial degree of reconstruction and d is the number of dimensions.

One of the professor mentioned that it is advised to use multigrid techniques for improving the CFL condition. I am using unstructured higher order FVM. My question is,

1. Is it really required to do a multigrid for a transient problem when higher order schemes (3rd to 4th order) are used?

2. If I get a stable time marching with CFL=0.5 with third order RK and third order scheme, does it compromise the solution accuracy for a transient problem?

3. For unstructured higher order finite volume method, which is the best suitable way to do a multigrid?

Thanks in advance for your help, Shyam

Re: CFL condition for higher order schemes
 

1. No, but you may need to utilize limiters on your slope or flux reconstructions. 2. Depends on the problem - stability does not ensure accuracy. If the time step is larger than the characteristic time of the physics you are modeling then you won't capture the physics. 3. In my opinion, algebraic multigrid works well and is relatively easy to implement.

Re: CFL condition for higher order schemes
 

Dear Shyam,

1. Multigrid is a convergence acceleration device which works on the basic principle that low frequency errors on a finer grid are seen as high frequency errors on the coarse gird, and thereby use of relaxation schemes on sequence of grids to dampen the errors. In a transient problem, multigrid would become an essential ingredient if you are using dual--time stepping and casting the unsteady problem as a pseudo steady problem. Multigrid would then be a strong tool for accelerating convergence to the steady state in the dual loop, with its ability to dampen both low and high frequency errors.

2. A fixed CFL means a fixed physical time step, which needs to be lesser than the timescale involved in the problem to resolve the physics, just as ag pointed out in his post. The temporal error is proportional to the cube of the timestep used in your case, but the error fall rate is 3, independent of the timestep employed. On a given grid, as long as the solution is stable, spatial accuracy remains unaffected.

3. I have no personal experience in implementing multigrid on unstructured solvers, but I believe AMG would be an ideal choice. You can have a look at the works of Venkatakrishnan and Mavriplis (also VKI lecture series) for more information on multigrid in an unstructured framework.

Hope this helps.

Regards,

Ganesh