7.2.2 Sequential solving

• Discretization in time via an explicit 2-level difference scheme, with $\vec{u}^K := \vec{u}(t_K)$ and $\tau=t_{K+1}-t_K$ :

  $$\boxed{ \begin{split}& \vec{u}^0 \qquad \text{known from inital... ...ad \forall K=0,1,\ldots \ [0.5ex] & \text{+ B.C.} \end{split} }$$ (7.11)

  
  

• We use a finite volume method with degrees of freedom $u_j^K := \vec{u}^K(x_j)$, $x_j \in T_j$ located in the center of gravity of the volumes $T_j$, $j=\overline{1,N} \makebox[0pt]{}$. This leads to the system of equations :

  $$\boxed{ \begin{split}u_j^0 &\;=\; \frac{1}{\vert T_j\vert} \int... ...ne{1,N} \makebox[0pt]{} \qquad \forall K=0,1,\ldots \end{split} }$$ (7.12)

  
  

• Application of Gauß' theorem changes the volume integral in (7.15) into an integral on the surface of that volume. This integral can be expressed as a sum of the ${n_e(j)}$ integrals on the edges/faces $S_{jl}$ of the volume element $T_j$. Therein, $n_{jl}$ is the outer normal vector of $S_{jl}$.
  $$\int_{T_j} \mathrm{div} f(\vec{u}^K) dx \;=\; \int_{\partial T_... ... \sum\limits_{l=1}^{n_e(j)} \int_{S_{jl}} f(\vec{u}^K(s)) \vec{n}_{jl} ds$$

  Figure 7.1: Notations in the volume element $T_j$.

  

• Besides the stability of the explicit scheme, the crucial point of the numerical solving (and its parallelization) is the approximation of flux on the faces  $\; \int_{S_{jl}} \cdot \;$ by a numerical flux

  $$g_{jl}(\underline{u}^K) \;\approx\; \int_{S_{jl}} f(\underline{u}^K(s)) \vec{n}_{jl} ds \enspace.$$ (7.13)

  
  
  Then the numerical scheme for solving (7.13) is

  $$\boxed{ \begin{split}u_j^0 &\;=\; \frac{1}{\vert T_j\vert} \int... ...ine{1,N} \makebox[0pt]{} \quad \forall K=0,1,\ldots \end{split} }$$ (7.14)

  
  
  A brief discussion on the calculation of $g_{jl}$ will be given in the following.

Flux vector splitting in the calculation of $g_{jl}$

We follow exactly the notation in Kröner [Krö97], p.374f.
In (7.13), we assume now

$$f_1(\vec{u}) \;=\; f'_1(\vec{u})\cdot\vec{u}$$   and$$\qquad f_2(\vec{u}) \;=\; f'_2(\vec{u})\cdot\vec{u} \enspace.$$

The definition

$$C_{jl}(\vec{u}) \;:=\; \vec{n}_{jl} f'(\vec{u})$$

can be expressed as $Q^{-1}C_{jl}Q = D = \mathrm{diag}\left\{\lambda_1,\ldots,\lambda_m \right\}$ , with a non-singular matrix   $Q(\vec{u})$. Splitting the eigenvalues $\lambda_i$ into a positive and a negative set $\lambda_i^{+} := \max\left\{ \lambda_i, 0 \right\}$, $\lambda_i^{-} := \min\left\{ \lambda_i, 0 \right\}$, $i=\overline{1,m} \makebox[0pt]{}\;$ we can distinguish between inflow and outflow on that edge/face :

$$C_{jl}^{+}(\vec{u}) = Q D^{+} Q^{-1}$$   und$$\qquad C_{jl}^{-}(\vec{u}) = Q D^{-} Q^{-1} \enspace.$$

Because of $\vec{n}_{jl} = - \vec{n}_{lj}$ the relation $C_{jl}(\vec{u}) = - C_{lj}(\vec{u})$ is also valid. Especially, $C_{jl}^{-}(\vec{u}) = - C_{lj}^{+}(\vec{u})$ holds.

Figure 7.2: Flux between volume elements $T_j$ and $T_l$ (Steger and Warming).

Now, we can calculate the numerical flux $g_{jl} := \int_{S_{jl}} \widetilde{g} \makebox[0pt]{}_{jl}(s)ds$ by the formula

$$g_{jl}(w,v) \;:=\; \vert S_{jl}\vert \left( C_1(w,v) w + C_2(w,v) v \right) \enspace.$$ (7.15)

There exist several approaches for the calculation of the values $C_1$ and $C_2$ :

1. Scheme by Steger and Warming [SW81]

   $$\begin{split}C_1(w,v) &\;:=\; C_{jl}^{+}(w) \ C_2(w,v) &\;:... ...stackrel{\mathrm{!}}{=}\; - C_{lj}^{+}(v) \;\right] \end{split}$$ (7.16)

   
   

2. Scheme by Vijayasundaram [Vij86]

   $$\begin{split}C_1(w,v) &\;:=\; C_{jl}^{+}(\frac{w+v}{2}) \ C... ...thrm{!}}{=}\; - C_{lj}^{+}(\frac{w+v}{2}) \;\right] \end{split}$$ (7.17)

   
   

3. Scheme by van Leer [Lee92]

   $$\begin{split}C_1(w,v) &\;:=\; C_{jl}(w) + \left\vert C_{jl}^{... ...vert C_{lj}^{+}(\frac{w+v}{2})\right\vert \;\right] \end{split}$$ (7.18)

   
   

Remark :
Matrices $C_{jl}^{+}(w)$, $C_{jl}^{-}(w)$ and $C_{jl}(w)$ can be calculated locally in $T_j$. Similarly, $C_{lj}^{+}(v)$, $C_{lj}^{-}(v)$ and $C_{lj}(v)$ can be calculated locally in $T_l$. If one applies the argument $\frac{w+v}{2}$ then the average value $\left(\sum\right)$] $\left(\sum\right)$ has to be determined previously.
Those formulas in (7.19)-(7.21) marked in brackets $[\quad]$ can calculate $C_2$ locally in $T_l$. In general, the relation $\underbrace{C_2(w,v)}_{T_j} = -\underbrace{C_1(v,w)}_{T_l}$ is valid.