[joern]

Hi,

I read the OpenFOAM Code parts for the TVD-Schemes and i have some questions about it.

In literature the TVD-schemes are derived in a time independend manner.
For a value U it is:

U_f = (U_f)_low + limiter(r)*((U_f)_high - (U_f)_low)

with (upwind direction)

r = (U_P - U_W)/(U_E - U_P)

In OpenFOAM r is calculated like Darwish and Moukalled recommend.


In my literature these terms are time independend. In OF the calculation is time dependend (clear), but the factor r is calculated explicit but it is used for an implicit fade.
In other words: r is calculated for a time t^n but it is used in a fade of (U_f)_high and (U_f)_low at the time t^(n+1).


Is this still consistent to literature? Why?
Why does this scheme still satisfy the TVD condition?

I suppose its still a TVD Scheme, because the limiter is time independed and still satisfy Swebys diagram.
Is this true?


I hope someone can help, because i need to explain the reason, why this way of calculation is consistent.

Thx.

[joern]

Can anyone help me?

i need help.

OF uses the TVD schemes in an implicit way, but the factor r for the limiter is calculated explicit.

why is this kind of scheme still TVD?
In literature i only find full explicit or full implicit schemes -.-

[santiagomarquezd]

Hello Joern, I started to study NVD/TVD limiters in FOAM recently, I have some comments and questions about this topic.

1. As you said r is calculated explicitly from values of U_i^n. Next you assemble time terms and solve the system for U_i^(n+1).

2. Why you propose, "In my literature these terms are time independent"? TVD criterion have to be accomplished every timestep, so in each timestep it is neccessary to recalculate r an then psi(r) to limite the flux. r is different across the time because the changes in U.

3. About this paragraph: "In OF the calculation is time dependend (clear), but the factor r is calculated explicit but it is used for an implicit fade.
In other words: r is calculated for a time t^n but it is used in a fade of (U_f)_high and (U_f)_low at the time t^(n+1)."
What is fading?

4. What literature do have? I used the book from Hirsch, Jasak's Thesis and original papers. What literature have been helpful for you?, specially in implicit TVD.

Regards.

[joern]

Originally Posted by santiagomarquezd

Hello Joern, I started to study NVD/TVD limiters in FOAM recently, I have some comments and questions about this topic.

1. As you said r is calculated explicitly from values of U_i^n. Next you assemble time terms and solve the system for U_i^(n+1).

not i do so, OF does.

Originally Posted by santiagomarquezd

2. Why you propose, "In my literature these terms are time independent"? TVD criterion have to be accomplished every timestep, so in each timestep it is neccessary to recalculate r an then psi(r) to limite the flux. r is different across the time because the changes in U.

Thats right, r has to be calculated every timestep but most literature, i found, either used a complete explicit TVD sheme or ignored the time index in their text.
When i worte that i just meant the syntax.

Originally Posted by santiagomarquezd

3. About this paragraph: "In OF the calculation is time dependend (clear), but the factor r is calculated explicit but it is used for an implicit fade.
In other words: r is calculated for a time t^n but it is used in a fade of (U_f)_high and (U_f)_low at the time t^(n+1)."
What is fading?

fade? thats what TVD does.
TVD does a mix of high and low schemes by using factors z and 1-z. So you can get values from full-high- to full-low-scheme.

Originally Posted by santiagomarquezd

4. What literature do have? I used the book from Hirsch, Jasak's Thesis and original papers. What literature have been helpful for you?, specially in implicit TVD.

Regards.

my literature was:
M. S. Darwish, F. Moukalled. TVD schemes for unstructured grids.
International Journal of Heat and Mass Transfer 46, S. 599611, 2003

J. H. Ferziger, M. Peric. Computational Methods for Fluid Dynamics.
Springer Verlag, 2. Au
age, 1997

P. K. Sweby. High resolution schemes using flux limiters for hyperbolic
conservation laws. SIAM J. Numer. Analysis, 21:995-1011,
1984

H. K. Versteeg, W. Malalasekera. An Introduction to Computational
Fluid Dynamics: the fnite volume method.
Pearson Edudacion Limited, 2. Aufage, 2007

and some other papers i think.

[santiagomarquezd]

Hello Joern, thanks for your reply,

fade? thats what TVD does.
TVD does a mix of high and low schemes by using factors z and 1-z. So you can get values from full-high- to full-low-scheme.

That's that I supposed, I think in FOAM framework it is called blending (HRV PhD Ts. chap. 3.4.3).

H. K. Versteeg, W. Malalasekera. An Introduction to Computational
Fluid Dynamics: the fnite volume method.
Pearson Edudacion Limited, 2. Aufage, 2007

Except for this book I have the literature you suggested. I've read previous edition, but Ola Widlund told me that it is improved now. Do you have an electronic copy of it (my mail is in the contact information page)?

Regards.

[joern]

Originally Posted by santiagomarquezd

Except for this book I have the literature you suggested. I've read previous edition, but Ola Widlund told me that it is improved now. Do you have an electronic copy of it (my mail is in the contact information page)?

Regards.

Sry, i have no electronic copy. i found the book in our bib but not in the part for math. i found it within books for mechanical engeneering.