[celestial]

These are described at the end of Chapter 3 of

Numerical Heat Transfer and Fluid Flow by Patankar, 1980

but I never see them mentioned in CFD books like for example

Computational Fluid Dynamics: Principles and Applications by Blazek

Numerical Computation of Internal and External Flows by Hirsch

Computational Gasdynamics by Laney

An Introduction to Computational Fluid Dynamics - the Finite Volume Method
by Versteeg and Malalasekera


Why is that so ?

[agd]

Rather than forcing your readers to dig up reference books that may not be quickly available, why not list the conditions you refer to? If you want an answer to a question try not to make your audience do all the work for you.

[celestial]

You have a point. Coming to think of it not everybody these days refers to a 45 year book. I will gladly list them:



Rule 1: Consistency at control-volume faces. When a face is common to
two adjacent control volumes, the flux across it must be represented by
the same expression in the discretization equations for the two control
volumes.

Rule 2: All coefficients in discretization equations must always be
of the same sign.

Rule 3: Negative-slope linearization of the source term. This is a must to avoid instabilities.

Rule 4: a coefficient equals the sum of the neighbor coefficients. This ensures that adding a constant to our unknown field variable satisfies the same differential equation.


I can attach a pdf of chapter 3 discretization methods if this is still unclear.

Kindest Regards

[LuckyTran]

Of course I am not here to harp on Patankar, it is a great text. But the rules are there for pedagogical purposes, not some grand theory.

Rule 1. If a=b then b=a. I admit I have never once cited in any publication the symmetric property even though I have known it since I was about 12 years old.
Rule 2. A positive semi-definite matrix can be obtained if all its elements are positive, a trivial case.
Rule 3. A negative definite matrix is not positive semi-definite. Also trivial.
Rule 4. It is preferably for the matrix to be diagonally dominant

Unironically, there is an algebraic theorem that says exactly all that:
A symmetric diagonally dominant matrix with real positive diagonal entries is positive definite

How come nobody is citing this theorem? That's not the point. The book is to attempting to explain how these results come about.

[aerosayan]

Also expanding on Rule-4: What Patankar is trying to say is ...

The common linear system equation , can also be represented as

Generally is used as

Why?

Because it makes the matrix diagonally dominant, so it's easier for iterative solvers to converge faster.

[arjun]

"Rule 4: a coefficient equals the sum of the neighbor coefficients. This ensures that adding a constant to our unknown field variable satisfies the same differential equation."


Would not converge in normal case. This type of matrix you get out of pure neumann case where diagonal is equal to negative sum of off diagonals.

You need at least one equation where diagonal is more than (not just equal to) negative sum of off dias.

[FMDenaro]

To be honest, such rules are at present only a mandatory task for CFD students no longer for scientific publications.

[NotOverUnderated]

Originally Posted by FMDenaro

To be honest, such rules are at present only a mandatory task for CFD students no longer for scientific publications.

But aren't those books targeting students?

[sbaffini]

Originally Posted by celestial

These are described at the end of Chapter 3 of

Numerical Heat Transfer and Fluid Flow by Patankar, 1980

but I never see them mentioned in CFD books like for example

Computational Fluid Dynamics: Principles and Applications by Blazek

Numerical Computation of Internal and External Flows by Hirsch

Computational Gasdynamics by Laney

An Introduction to Computational Fluid Dynamics - the Finite Volume Method
by Versteeg and Malalasekera


Why is that so ?

Can't speak for all the books, but I'm pretty sure they are, maybe just not in the same pedagogical form or order.

[LuckyTran]

We regularly break these rules all the time. Especially rules 2, 3, and 4.


These rules are like speed limits signs on roadways. They are suggestions for what ought to be done but it it's frequently necessary to break the rule. You have to drive faster than the speed limit so you don't get rear-ended by driver behind or when you need to overtake someone that is doing the speed limit. So when you teach someone to drive, do you tell them to never, ever, ever go faster than the speed limit?

[celestial]

In chapter 3, Patankar talks about the physics of the 1D thermal diffusion equation with source.

At this stage in the text, he is not concerned about the numerical aspects to the solution of Ax=b (for example, whether matrix A is orthogonal, symmetric definite or whatever).

He refers to these rules throughout the book (even to the incompressible NavierStokes equations).

If you are interested in reading it online, here is the link to the whole book:

https://ia902208.us.archive.org/13/i...ankar_text.pdf

At any rate, great thanks to all of you for the replies.

[arjun]

I am very sure everyone commenting here has read patankar's book.

[LuckyTran]

The book is titled Numerical Heat Transfer and Fluid Flow. I want to say it's on the first page but it's actually on the cover!

[FMDenaro]

Originally Posted by celestial

In chapter 3, Patankar talks about the physics of the 1D thermal diffusion equation with source.

At this stage in the text, he is not concerned about the numerical aspects to the solution of Ax=b (for example, whether matrix A is orthogonal, symmetric definite or whatever).

He refers to these rules throughout the book (even to the incompressible NavierStokes equations).

If you are interested in reading it online, here is the link to the whole book:

https://ia902208.us.archive.org/13/i...ankar_text.pdf

At any rate, great thanks to all of you for the replies.

I don't know if you have read the CFD textbooks, if you had you would have known that rule 1) is largely discussed and is a fundamental property of FVM.