CFL Condition Implicit Method

Velocimetry · June 29, 2018, 19:03

An implicit method is unconditionally stable. Why would one have a CFL condition when solving with an implicit method?

Is there another type of error, that is being obtained from PDE's that I am overlooking?

I understand, it resolves the time dependent features one is looking for. So, for the sake of the question, if one does not care to resolve the time dependent features, why would one need proper CFL condition for solver stability in an Implicit Method.

FMDenaro · June 30, 2018, 03:14

Quote:

Originally Posted by Velocimetry

An implicit method is unconditionally stable. Why would one have a CFL condition when solving with an implicit method?

Is there another type of error, that is being obtained from PDE's that I am overlooking?

I understand, it resolves the time dependent features one is looking for. So, for the sake of the question, if one does not care to resolve the time dependent features, why would one need proper CFL condition for solver stability in an Implicit Method.

It is an accuracy issue...if you look at the local truncation error you will see the CFL. In a more physical words, the cfl is the ratio between the physical velocity and dx/dt that is a numerical advection velocity allowed by the cell dimension and time step. TO be accurate the transient simulation this rate has to be no more than 1.

sbaffini · June 30, 2018, 09:23

This might be obvious but, stability is never actually "unconditional". There always are conditions, besides the CFL numerical value, under which you can prove an implicit method is stable. Sometimes they are very loose, sometimes more stringent.

For example, most dimonstrations make use of uniform cartesian grids. Certain grid features can make unstable an otherwise stable method.

Also, stability is typically proven for a given method on a given equation. For a number of reasons you could not be able to prove/have stability on a different equation.

Besides this, of course, there are accuracy concerns.

June 29, 2018, 19:03	CFL Condition Implicit Method	#1
Velocimetry New Member Join Date: Mar 2018 Posts: 14 Rep Power: 8	An implicit method is unconditionally stable. Why would one have a CFL condition when solving with an implicit method? Is there another type of error, that is being obtained from PDE's that I am overlooking? I understand, it resolves the time dependent features one is looking for. So, for the sake of the question, if one does not care to resolve the time dependent features, why would one need proper CFL condition for solver stability in an Implicit Method.

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June 30, 2018, 09:23		#3
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,151 Blog Entries: 29 Rep Power: 39	This might be obvious but, stability is never actually "unconditional". There always are conditions, besides the CFL numerical value, under which you can prove an implicit method is stable. Sometimes they are very loose, sometimes more stringent. For example, most dimonstrations make use of uniform cartesian grids. Certain grid features can make unstable an otherwise stable method. Also, stability is typically proven for a given method on a given equation. For a number of reasons you could not be able to prove/have stability on a different equation. Besides this, of course, there are accuracy concerns.