Research Group of Prof. Dr. M. Griebel
Institute for Numerical Simulation
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Discretization of boundary conditions

As described in [1], Dirichlet conditions for the velocity are implemented by setting the nodal value of the normal velocity component or by linear interpolation for the tangential velocity components. The homogeneous boundary Neumann conditions for the pressure are discretized by, e.g.

$\displaystyle p_{0,j,k}=-p_{1,j,k}~.

In case of convex edges some special treatment is required. Consider the situation shown in Figure [*].
Figure: convex corner
If $ \Gamma_1$ denotes the face between the cells $ _{i,j}$ and $ _{i,j+1}$ and $ \Gamma_2$ the face between the cells $ _{i,j}$ and $ _{i+1,j}$, then the homogeneous Neumann conditions are discretized by

$\displaystyle p_{i,j}=-\frac{\vert\Gamma_1\vert p_{i,j+1}+\vert\Gamma_2\vert p_{i+1,j}}{\vert\Gamma_1\vert+\vert\Gamma_2\vert} \quad.

Martin Engel 2004-03-15