# Fromm based Schemes - structured grids

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## Contents

### 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

 $\hat{\phi_{f}}= \begin{cases} 2 \hat{\phi_{C}} & 0 \leq \hat{\phi_{C}} \leq \frac{1}{4} \\ \frac{1}{4} + \hat{\phi_{C}} & \frac{1}{4} \leq \hat{\phi_{C}} \leq \frac{3}{4} \\ 1 & \frac{3}{4} \leq \hat{\phi_{C}} \leq 1 \\ \widehat{\phi_{C}} & \widehat{\phi_{C}} \triangleleft 0 \ , \ \widehat{\phi_{C}} \triangleright 1 \end{cases}$ (2)

Normalized variables - non-uniform grids

 $\hat{\phi_{f}}= \begin{cases} 2 \hat{\phi_{C}} & 0 \leq \hat{\phi_{C}} \leq x_{Q}/2 \\ a_{f} + b_{f} \hat{\phi_{C}} & x_{Q}/2 \leq \hat{\phi_{C}} \leq 3 x_{Q}/2 \\ 1 & 3 x_{Q}/2 \leq \hat{\phi_{C}} \leq 1 \\ \hat{\phi_{C}} & \hat{\phi_{C}} \triangleleft 0 \ , \ \hat{\phi_{C}} \triangleright 1 \end{cases}$ (2)

where

 $a_{f}= \left( 3 x_{Q} - 2 \right)/2$ (2)
 $b_{f}= \left( 1 - x_{Q} \right) / x_{Q}$ (2)

### van Leer limiter

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