Flux computation in unstructured grids

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Flux computation in unstructured grids

Posted July 16, 2010 at 11:49 by praveen

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

$Q^{n+1}_j = Q^n_j - \Delta t \sum_{k \in N(j)} F(Q_j^n, Q_k^n, n_{jk})$

Here $n_{jk}$ 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

$F(U,V,n) = \frac{1}{2}[ F(U) \cdot n + F(V) \cdot n] - \frac{1}{2} \lambda(U,V,n) (V - U)$

Here, we have used the definition

$F(U) \cdot n = \begin{bmatrix} \rho (u \cdot n)\\ \rho u_x (u \cdot n) + p n_x\\ \rho u_y (u \cdot n) + p n_y\\ \rho u_z (u \cdot n) + p n_z\\ (E+p)(u \cdot n) \end{bmatrix}$

where