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Unformatted text preview: se its error is proportional to
rather than .
Now we consider the equation we want to solve: () () () () We assume a uniform grid of the following form Where , and We will initially assume consistent cross sections throughout the mesh. This simplifies matters
considerable. An alternative is to require constant cross-sections within each cell, but allow the cross
sections to vary between cells. This complicates matters, because we must build the interface
conditions into the scheme. Since we are developing a finite volume scheme, we obtain the basic equation for a single cell by
integrating the diffusion equation over the cell;
∫ ∫ This balance equation suggests that we have scalar fluxes at cell centers and currents at cell edges. The
above equation is actually exact, but now we introduce an approximation through the cell-centered a
flux. Next we further introduce approximations by using central-difference approximations for the currents:
( ) Substituting this expression into our approximate balance equation, we obtain our interior-mesh
( ) ( ) Now, we must derive a differe...
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- Fall '10