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Symmetric boundary condition

In some of the simulations, we can assume a symmetric state exists on the boundary. This treatment of the boundary condition corresponds to the physical assumption that, on the two sides of boundary, the same physical processes exist. The variable values at the same distance from the boundary at the two sides are the same. The function of such a boundary is that of a mirror that can reflect all the fluctuations generated by the simulation region.

Assume f(xb), \(f(x_b-\Delta x)\) are two adjacent boundary values, a typical set of symmetric boundary value is

\begin{displaymath}f(x_b+\Delta x)=f(x_b)
\end{displaymath} (57)

\begin{displaymath}f(x_b+2\Delta x)=f(x_b-\Delta x)
\end{displaymath} (58)