Fromm based Schemes - structured grids

=== Fromm scheme ===

J.E.Fromm

A method for reducing dispersion in convective difference schemes

J. Comp. Phys., Vol. 3, p.176, (1968)

=== MUSCL - Monotonic Upwind Scheme for Conservation Laws ===

'''Lien F.S. and Leschziner M.A.''' , Proc. 5th Int. IAHR Symp. on Refind Flow Modelling and Turbulence Measurements, Paris, Sept. 1993

Based on Fromm's scheme

 
Normalized variables - uniform grids

<table width="100%"><tr><td>
:<math> 
\hat{\phi_{f}}=  
\begin{cases}
2 \hat{\phi_{C}}                  &  0          \leq \hat{\phi_{C}} \leq \frac{1}{4} \\ 
\frac{1}{4} + \hat{\phi_{C}}      & \frac{1}{4} \leq \hat{\phi_{C}} \leq \frac{3}{4} \\
1                                 & \frac{3}{4} \leq \hat{\phi_{C}} \leq 1 \\    
\widehat{\phi_{C}} & \widehat{\phi_{C}} \triangleleft 0 \ , \ \widehat{\phi_{C}} \triangleright 1
\end{cases}
</math>
</td><td width="5%">(2)</td></tr></table>

Normalized variables - non-uniform grids

<table width="100%"><tr><td>
:<math> 
\hat{\phi_{f}}=  
\begin{cases}
2 \hat{\phi_{C}}              &  0        \leq \hat{\phi_{C}} \leq x_{Q}/2   \\ 
a_{f} + b_{f} \hat{\phi_{C}}  & x_{Q}/2   \leq \hat{\phi_{C}} \leq 3 x_{Q}/2 \\
1                             & 3 x_{Q}/2 \leq \hat{\phi_{C}} \leq 1         \\    
\hat{\phi_{C}} & \hat{\phi_{C}} \triangleleft 0 \ , \ \hat{\phi_{C}} \triangleright 1
\end{cases}
</math>
</td><td width="5%">(2)</td></tr></table>

where

<table width="100%"><tr><td>
:<math>
a_{f}=  \left( 3 x_{Q} - 2 \right)/2
</math>
</td><td width="5%">(2)</td></tr></table>

<table width="100%"><tr><td>
:<math> 
b_{f}= \left( 1 - x_{Q} \right) / x_{Q} 
</math>
</td><td width="5%">(2)</td></tr></table>

=== van Leer limiter ===

=== van Albada ===

Bounded Fromm

G.D. Van Albada, B.Van Leer, W.W.Roberts

A comparative study of computational methods in cosmic gas dynamics

Astron. Astrophysics, Vol. 108, p.76, 1982

=== OSPRE ===

bounded Fromm

Waterson [1995]

N.P.Waterson, H.Deconinck.

A unified approach to the design and application of bounded high-order convection schemes

In C. Taylor and P.Durbetaki, editors, Proc. Ninth Int. Conf. on Numer. Method. Laminar and turbulent Flow, pages 203-214, Pineride Press, Swansea, 1995

---------------------------------------------------------------------
----
<i> Return to [[Numerical methods | Numerical Methods]] </i>

<i> Return to [[Approximation Schemes for convective term - structured grids]] </i>