3.4.2 Fluxes



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3.4.2 Fluxes

 

Our expressions derived for the transformed differential operators are the result of a box integration over subdomains . These subdomains partition the interior of the simulation domain without overlap or exclusion. It is quite obvious to use subdomains (3.4-7) that split the domain at the mid interval lines of the grid (Figure 3.4-3), which are identical to the Voronoi cells (cf. Figure 3.3-1).

 

 

We use the common notation for function values at mid interval points (3.4-8), (3.4-9) and the grid spacing in the computational domain (3.4-10).

 

 

 

In the following we repeatedly require discrete approximations for partial derivatives of a function . If is sufficiently smooth (here three times differentiable), we may use the difference expression (3.4-11) for the partial derivatives. The local truncation error is the residual which occurs when inserting the solution of the continuous problem into the discrete scheme. The local truncation error in our approximation (3.4-11) is of first order in the grid spacing weighted with the second partial derivatives (3.4-12). Therefore, we call the approximation (3.4-11) first order accurate.

 

 

The most crucial and computationally expensive task is to find an appropriate discrete representation for the flux components. To avoid unnecessary complications we look closely at just one term of the sum in (3.1-2), at the partial flux of quantity which is related to (or driven by) quantity . The transformation of the equations from physical space to computational space is linear and, therefore, we can merely consider the term

 

and then discretize its transformed counterpart and accumulate the discretized fluxes, without loss of generality. The term is not explicitly indicated in the following derivations. It is treated as an independent additional contribution to the flux .

The transformed flux is given by (3.4-14) and (3.4-15), which is obtained by applying (3.3-55) to (3.4-13).

 

 

For the transformed expression of the divergence (3.4-16), we need both flux components and at all mid interval points and . These points are marked by in Figure 3.4-3.

 

For the discretization of these flux components and at a mid interval point we use the local vertical six point stencil shown in Figure 3.4-4. We denote values at the fictive center point () with an index and the neighbor points () which are actual grid points with indices 1 to 6. For the mid interval point we use the analogous horizontal six point stencil.

 

For numerical reasons we distinguish between the cases where we have

-
only a diffusive flux,
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only a convective flux,
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diffusive and convective flux.





next up previous contents
Next: Diffusion Fluxes Up: 3.4 Discretization of the Previous: 3.4.1 Divided Differences



Martin Stiftinger
Wed Oct 19 13:03:34 MET 1994