boundedness property

ysh1227 · June 19, 2012, 12:20

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.

cfdnewbie · June 19, 2012, 16:37

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

sbaffini · June 19, 2012, 17:16

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.

cfdnewbie · June 19, 2012, 17:30

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.

ysh1227 · June 19, 2012, 23:44

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.

michujo · June 20, 2012, 03:40

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.

ysh1227 · June 20, 2012, 04:22

thanks, michujo.

FMDenaro · June 20, 2012, 05:50

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.

June 19, 2012, 12:20	boundedness property	#1
ysh1227 New Member Seon Hye Join Date: Feb 2012 Location: South Korea Posts: 18 Rep Power: 14	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. ahparvin likes this.

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June 19, 2012, 16:37		#2
cfdnewbie Senior Member cfdnewbie Join Date: Mar 2010 Posts: 557 Rep Power: 20	it seems that the solution stays bounded (i.e. no infinity) in time, so your code does not blow up due to unphysical values.

June 19, 2012, 17:16		#3
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,195 Blog Entries: 29 Rep Power: 39	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.

June 19, 2012, 17:30		#4
cfdnewbie Senior Member cfdnewbie Join Date: Mar 2010 Posts: 557 Rep Power: 20	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.

June 19, 2012, 23:44		#5
ysh1227 New Member Seon Hye Join Date: Feb 2012 Location: South Korea Posts: 18 Rep Power: 14	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.

June 20, 2012, 03:40		#6
michujo Senior Member Join Date: Dec 2011 Location: Madrid, Spain Posts: 134 Rep Power: 16	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.

June 20, 2012, 04:22		#7
ysh1227 New Member Seon Hye Join Date: Feb 2012 Location: South Korea Posts: 18 Rep Power: 14	thanks, michujo.

June 20, 2012, 05:50		#8
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,896 Rep Power: 73	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.