From: Michael Buck <buck@ike.uni-stuttgart.de>
Subject: Re: TVD advection problem
Date: Wed, 19 Jan 2000 17:36:51 +0100
Newsgroups: sci.math.num-analysis
Summary: [missing]

I recommend you take a look into

C. Hirsch, Numerical Computation of Internal and External Flows, Vol 2, Chapter
21.4.4, Wiley and Sons, ISBN 0471 92351 6

or

H.C. Yee, Construction of explicit and implicit symmetric TVD schemes and their
applications, J. Computational Physics, 68, 151-179

According to these, the first order implicit Euler method (I think this is what
you refer by UD in the time domaine) for the time discretization together with a
appropriate TVD space discretization is unconditionally stable AND
unconditionally TVD (as you observed). For a second order implicit method such
as Crank-Nicholson the TVD property is subject to restrictions by the Courant
number CFL = |velocity| * delta t / delta x, i.e. for a sufficiently large
timestep, the TVD properties are lost.

I do not know what you mean by unacceptable diffusion with the first order
method. If you are only interested in steady state solutions and the diffusion
is in the x domaine, I suspect that the dispersion of your TVD method depends on
the CFL number. If so, see the above cited references for better methods.

If you are interested in time-accurate solutions and the "diffusion" is in the
time domaine, I can only suggest that you implement a control of the local
discretization error of your time integration (e.g. extrapolation or embedded
methods of different order, see textbooks on ordinary differential equations)
and switch to second order (with the appropriate time step size for assuring
TVD) if the truncation error of the first order method becomes too large.

You could also post your question to sci.physics.computational.fluid-dynamics.
The people there may be more familiar with your problem.

Good luck,

Michael

Jason Bowsher wrote:

> I am trying to solve an advection problem which is proving rather difficult.
> I have a scheme which uses a TVD method in the X (size) domain which blends
> between a UD and CD scheme in the X domain.  This seems to work OK in the X
> direction.
>
> My problem is in the time domain.  I effectivly have elements which are
> elongated in the time domain and if delta T is large then a crank-nicolson
> scheme introduces unacceptable oscillations.  If I use a UD scheme in the
> time domain I get unconditional stability but diffusion is unacceptable.
>
> I would like to use a TVD type appraoch to the time domain in which I switch
> or blend between UD and crank-nicholson to avoid oscilations.  I can see
> that the switching or blending function must be in some way related to the
> element dimensions (delta X and delta T) and the advection velocity.
>
> Can anone tell me what the relation ship between these variables would be in
> order to produce a function for theta that avoids oscilations in regions of
> large gradients in the time domain.
>
> Thanks
>
> Jason