Flux computation in unstructured grids
Posted July 16, 2010 at 10:49 by praveen
Tags finite volume, unstructured
Consider finite volume scheme on unstructured grids for the Euler equations.
Let
be normal to a cell face and whose magnitude is equal to face area. Let
be the conserved vector. The finite volume update equation using forward Euler time discretization is
![Q^{n+1}_j = Q^n_j - \Delta t \sum_{k \in N(j)} F(Q_j^n, Q_k^n, n_{jk}) Q^{n+1}_j = Q^n_j - \Delta t \sum_{k \in N(j)} F(Q_j^n, Q_k^n, n_{jk})](/Forums/vbLatex/img/2350c073e162ab99f1a5ac985e3bf408-1.gif)
Here
is a normal vector pointing from current cell "j" into the neighbouring cell "k". Note that the conserved variable Q is updated in the global Cartesian coordinate frame.
As an example, the Rusanov flux would be defined as
![F(U,V,n) = \frac{1}{2}[ F(U) \cdot n + F(V) \cdot n] - \frac{1}{2} \lambda(U,V,n) (V - U) F(U,V,n) = \frac{1}{2}[ F(U) \cdot n + F(V) \cdot n] - \frac{1}{2} \lambda(U,V,n) (V - U)](/Forums/vbLatex/img/afbf757e031082087c6c22031139a5e1-1.gif)
Here, we have used the definition
![F(U) \cdot n = \begin{bmatrix}
\rho (u \cdot n)\\
\rho u_x (u \cdot n) + p n_x\\
\rho u_y (u \cdot n) + p n_y\\
\rho u_z (u \cdot n) + p n_z\\
(E+p)(u \cdot n)
\end{bmatrix} F(U) \cdot n = \begin{bmatrix}
\rho (u \cdot n)\\
\rho u_x (u \cdot n) + p n_x\\
\rho u_y (u \cdot n) + p n_y\\
\rho u_z (u \cdot n) + p n_z\\
(E+p)(u \cdot n)
\end{bmatrix}](/Forums/vbLatex/img/bee89383c20d19c10d28a429120bbf1d-1.gif)
where
is the velocity vector, etc., and
![\lambda(U,V,n) = \max\{ |u(U) \cdot n| + a(U) |n|, |u(V) \cdot n| + a(V) |n| \} \lambda(U,V,n) = \max\{ |u(U) \cdot n| + a(U) |n|, |u(V) \cdot n| + a(V) |n| \}](/Forums/vbLatex/img/3c1da7dd2161fe174f0cf27e74013311-1.gif)
with
being speed of sound. Note that there are many other ways to define
.
Let
![n=(n_x,n_y,n_z) n=(n_x,n_y,n_z)](/Forums/vbLatex/img/4a26e6508b60f710c7f410d9682c5cc3-1.gif)
![Q Q](/Forums/vbLatex/img/f09564c9ca56850d4cd6b3319e541aee-1.gif)
![Q^{n+1}_j = Q^n_j - \Delta t \sum_{k \in N(j)} F(Q_j^n, Q_k^n, n_{jk}) Q^{n+1}_j = Q^n_j - \Delta t \sum_{k \in N(j)} F(Q_j^n, Q_k^n, n_{jk})](/Forums/vbLatex/img/2350c073e162ab99f1a5ac985e3bf408-1.gif)
Here
![n_{jk} n_{jk}](/Forums/vbLatex/img/ed7a75fa0aef1539fe8d1d6bc2cc151e-1.gif)
As an example, the Rusanov flux would be defined as
![F(U,V,n) = \frac{1}{2}[ F(U) \cdot n + F(V) \cdot n] - \frac{1}{2} \lambda(U,V,n) (V - U) F(U,V,n) = \frac{1}{2}[ F(U) \cdot n + F(V) \cdot n] - \frac{1}{2} \lambda(U,V,n) (V - U)](/Forums/vbLatex/img/afbf757e031082087c6c22031139a5e1-1.gif)
Here, we have used the definition
![F(U) \cdot n = \begin{bmatrix}
\rho (u \cdot n)\\
\rho u_x (u \cdot n) + p n_x\\
\rho u_y (u \cdot n) + p n_y\\
\rho u_z (u \cdot n) + p n_z\\
(E+p)(u \cdot n)
\end{bmatrix} F(U) \cdot n = \begin{bmatrix}
\rho (u \cdot n)\\
\rho u_x (u \cdot n) + p n_x\\
\rho u_y (u \cdot n) + p n_y\\
\rho u_z (u \cdot n) + p n_z\\
(E+p)(u \cdot n)
\end{bmatrix}](/Forums/vbLatex/img/bee89383c20d19c10d28a429120bbf1d-1.gif)
where
![u=(u_x,u_y,u_z) u=(u_x,u_y,u_z)](/Forums/vbLatex/img/0185802fdb36a673457c84e3a426284c-1.gif)
![\lambda(U,V,n) = \max\{ |u(U) \cdot n| + a(U) |n|, |u(V) \cdot n| + a(V) |n| \} \lambda(U,V,n) = \max\{ |u(U) \cdot n| + a(U) |n|, |u(V) \cdot n| + a(V) |n| \}](/Forums/vbLatex/img/3c1da7dd2161fe174f0cf27e74013311-1.gif)
with
![a a](/Forums/vbLatex/img/0cc175b9c0f1b6a831c399e269772661-1.gif)
![\lambda \lambda](/Forums/vbLatex/img/c6a6eb61fd9c6c913da73b3642ca147d-1.gif)
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