boundedness property
 

Hi, guys.
I'm new in CFD and I have one question.
When I read the OpenFOAM programmer's guide, I found 'boundedness'.

For example,

1. Euler implicit uses implicit discretisation of the spatial terms, thereby taking current values.
It is forst order accurate in time, guarantees boundedness and is uncontionally stable.
2. Crank-nicholson uses the trapezoid rule to discretise the spatial terms, thereby taking a mean of current values and old values.
It is second order accurate in time, is unconditionally stable but does not guarantee boundedness.


I understood above all except about boundedness.
I know that this is basic question in CFD, but I can not understand that meaning exactly.
Could you explain about 'boundedness'? plz. :)

it seems that the solution stays bounded (i.e. no infinity) in time, so your code does not blow up due to unphysical values.

To make it more clear, imagine having a variable density flow or the transport of some artificial scalar (e.g., vof, progress variable, mixture fraction). Given a physically sound field at some time, independently from the accuracy of your scheme, you have to be sure that your integration of the equations will produce a physically sound distribution at the next time step (e.g., none of the above scalars can become negative). As a consequence, to avoid things going bad, you need a bounded scheme to integrate your equation in time.

This, of course, is the fast-easy explanation (the only one i'm able to give). There is a pretty huge mathematical basis now on boundedness, which can be found in several books.

or to offer another perspective: boundedness is usually a property of the scheme that prevents your numerical crimes (underresolution, wrong model) etc from biting you in the behind.

Another name for this concept would be TVB or TVD schemes.

Thanks, everyone.
Your comments are helpful for me.

And could I ask you one more thing?

What is the hypernyms of boundedness that could help to find the reference?
Just I want to know more about it.

Hi, you could have a look at the book "Numerical computation of internal and external flows" from Charles Hirsch, where some pages are devoted entirely to this concept.

Cheers.

thanks, michujo. :)

The starting observation is the Godunov theorem: Any linear monotone discretization of a PDE can be only first order accurate.
For example, solving
df/dt +c df/dx=0 with c=constant
with bounded initial data on f (e.g. [0,1]]), has a monotone solution only if you use a FTUS scheme, for any other linear discretization you are sure that new minimum and maximum extrema are created. To circumvent the Godunov theorem you must use non-linear schemes also for this linear PDE.
Many books treat the theoretical issue about monotone schemes and TVD property.