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Boundary conditions

  Following [25], we demonstrate our boundary treatment in the case of the left-boundary (see Figure 1.4.10).

Figure 1.4.10: Two dimensions - left boundary 

We distinguish between inflow (tex2html_wrap_inline5703), and outflow (tex2html_wrap_inline5705), boundary cells.

In inflow boundary cells, we reconstruct a constant interpolant from the prescribed point-values at these boundaries,
This reconstruction is then used to build the approximate solution at time tex2html_wrap_inline5707 in the interior cells. At the next-time step, tex2html_wrap_inline5707, the cell-averages at these boundary cells are defined according to the prescribed point-values as

We now turn to the outflow boundary cells. Here, we extrapolate the data from the interior of the domain, up to the boundary. First, we determine the discrete slope in the x-direction, tex2html_wrap_inline5713. This slope is then used to extrapolate the cell-average up to the boundary,
which is then used to predict the mid-value, tex2html_wrap_inline5715. Here
The discrete slope in the y-direction, tex2html_wrap_inline5719, is computed in that boundary cell in an analogous way to the interior computation. In summary, the staggered average at time tex2html_wrap_inline5707 is given by
This concludes the boundary treatment of the left boundary. Similar expressions hold for the other three boundaries.

We now turn to the corners and as a prototype, consider the upper-left corner (see Figure 1.4.11). In the corner we repeat the previous boundary treatment with one simple modification. The main difference regarding the boundary scheme in the corner is based on the number of different possible inflow/outflow configurations in that corner.

Figure 1.4.11: Upper-left corner 

Computationally, the most complicated case is when the flow in that upper-left corner is outflow in both directions. In this case, the staggered average at time tex2html_wrap_inline5707, tex2html_wrap_inline5725, is computed according to



The cell-average in the north-west edge of Figure 1.4.11 in time tex2html_wrap_inline5707, is given in this outflow-outflow case by the corrector step
When one of the boundaries is inflow, we have tex2html_wrap_inline5729, and tex2html_wrap_inline5731 (- the prescribed pointvalues at the corner).

As an example, we we approximate a solution to the two-dimensional Burgers equation
subject to the initial conditions,

and augmented with boundary conditions at the inflow boundaries which are equal to the initial values at these same boundaries. Figures 1.4.12 show the evolution of the solution in time for mesh sizes 41*41 and 81*81. Again, we note that there are no spurious oscillations at the boundaries, oscillations that are inherent with a naive treatment of inflow boundaries.

Figure 1.4.12: The 2D IBVP Burgers equation: T=1. (a) N=41, (a') N=81

next up previous contents
Next: Incompressible Euler equations Up: Central schemes in two Previous: Central schemes in two

Eitan Tadmor
Mon Dec 8 17:34:34 PST 1997