how implicit is implicit?
 

Hi, just a general question:

Suppose I have a time-dependent partial differential equation and would like to discretise (temporally) it into an implicit form. Does that mean that the time step size that I choose will not affected by the grid size that I choose to discretise my spatial derivatives?

Or will I also have to consider the method that I choose to solve my implicit form? Suppose, instead of solving simultaneously all the unknowns under one large matrix, I choose an iterative method, like a multiple predictor-corrector, will my time step size be more restricted then(as like in an explicit form of temporal discretisation)?

Re: how implicit is implicit?
 

Even if your disretization is implicit, if the solver is iterative, then you have time step size restriction. You can estimate that with local mode analysis.

Junseok

Re: how implicit is implicit?
 

How can this 'local mode analysis' be done?

Re: how implicit is implicit?
 

You can check most numerical analysis text books.

Junseok

Re: how implicit is implicit?
 

Hi, I guess the highest speed of local disturbance is critical. I think the delta_x/delta_t of the computational domain must be smaller than the physical speed of disturbances. So for the same grid size, time step has to be smaller for a shock wave of say Mach 3 then the Mach 1.5 wave. You can verify this with any one dimensional moving shock problem. -amol

Re: how implicit is implicit?
 

Ok, great, thanks all!

I'm trying to sort this out.. so I should say that even though the scheme is implicit, it really has to depend on how the scheme is solved. But I suppose the time size restriction has to be looser than that of an explicit scheme, right?

Can anyone verify with me if a Crank Nicolson 2nd order time discretization is a semi-implicit, semi-explicit scheme?

And, amol, I'm doing an incompressible flow of low to medium Re. So for this local speed, I should be using the maximum |(U,V)| within the domain, right? I suppose this is basically the CFL criterion.

Re: how implicit is implicit?
 

Usually we say Crank Nichoson is semi-implicit.

Re: how implicit is implicit?
 

yeah i think you are right Joe. -amol