TVD Schemes in OF
Hi,
I read the OpenFOAM Code parts for the TVDSchemes and i have some questions about it. In literature the TVDschemes 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. 
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 . 
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. 
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When i worte that i just meant the syntax. Quote:
TVD does a mix of high and low schemes by using factors z and 1z. So you can get values from fullhigh to fulllowscheme. Quote:
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, 21:9951011, 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. 
Hello Joern, thanks for your reply,
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Regards. 
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