Fromm's is C.F.D. scheme

$$u_j^{n+1}=u_j^n+\frac{\Delta t} {\Delta x}[F_{j-1/2}^{n+1/2}-F_{j+1/2}^{n+1/2}]$$

where

$$F_{j+1/2}=a(u_j^n-1/2*(1-\sigma)\Delta u_j)$$

where $\Delta u_j/\Delta x$ is the slope of the line segment in j^th cell and $\Delta u_j=(u_{j+1}^n-u_{j-1}^n)/2$

Slope limiter is a slope limited method, we choose $\Delta u_j$ depending on the nearby values of u.

Idea: Don't allow the interpolation function

$$u_I(x)=u_j^n+(x-x_j)/\Delta x *\Delta \widetilde{u_j}$$

to introduce new MAX or MIN's in the discrete data

$$U^n=(u_0,\cdots,u_{M-1})$$

we can do this by evaluationg u_I(x) at the edges of the jth cell x_j-1/2 and x_j+1/2 and compare with u_j-1,u_j,u_j+1:

for eg.

$$if u_I(x_{j+1/2})=u_j^n+1/2\Delta u_j >u_j,u_{j+1}$$

(where $\Delta u_j=(u_{j+1}-u_{j-1})/2$,then we set $\Delta \widetilde{u_j}=0$consider all possible slopes you can get from u_j-1,u_j,u_j+1 :

$$\Delta u_j^L=u_j-u_{j-1}\qquad \Delta u_j^R=u_{j+1}-u_j\qquad \Delta u^C=(u_{j+1}-u_{j-1})/2$$

(when you use $\Delta u_j=0$ in Fromm's method you get first order upwind.)

The Van Leer solpes $\Delta u_j^{VL}$ is given by:

$$\Delta^{VL}u_j= \left \{ \begin {array} {r c} S_j \cdot min(2... ..., if \quad lopes\quad change\quad sign \end {array} \right .$$

where $\phi =(u^n_{j+1}-u_j^n)(u_j^n-u^n_{j-1})$

S_j=sign(u_j+1-u_j-1)

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The translation was initiated by Wenlong Jin on 1999-02-10