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\noindent
\hbox to \textwidth{\footnotesize Proc. of 7th Conference of China Numerical Methods 
		on Fluid Mechanics	\hfill}
\hbox to \textwidth{\footnotesize Changchun, China, 1995, p.50 	\hfill}
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\Large 
On the Down Flow Boundary, 
an Extrapolation that Preserves TVD for Out Flow Problems\footnote{
Original title is: ``An Extrapolation that Preserves TVD
 for Out Flow Problems''}
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\normalsize YU Xin\footnote{E-mail: yu@cerse.psu.edu \quad

~~Home pages: http://www.cerse.psu.edu/yu/, http://www.yuxin.com/}, WANG Famin, ZHAO Lie\\ 
%(Institue of Mechanics, Chinese Academy of Sciences, Beijing 100080, China)
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(Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, China)
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{\large [First paragraph]} ---- omitted.	\vskip2em

When we solve an out flow problem numerically,
we cut the infinite solution domain to a finite scale.
One method to decide the value on outflow boundary is extrapolation.
Zero gradient extrapolation has a good stability with low order accurate;
Higher order accurate extrapolations sometimes lose stability.
Now we present an extrapolation (on the boundary) 
that Preserves TVD for out flow problems. 
It obtains both higher order accurate and a good stability.
It can be extended to other schemes.

For a three point TVD scheme, Set $U_N^{n+1}$ on the boundary as
$$
U_N^{n+1}=U_{N-1}^{n+1}+{\rm sign}(\Delta_{N-\frac 12}^{n+1}U){\rm min}
(|\Delta_{N-\frac 12}^{n+1}U|,|U_N^n-U_{N-1}^{n+1}|)	\eqno{(1)}
$$
where $\Delta_{N-\frac 12}^{n+1}U$ is a first order accurately (linearly)
 extrapolated value
minus $U^{n+1}_{N-1}$, or a second order accurately extrapolated value
minus $U^{n+1}_{N-1}$, or\\ ``min\,mod''\footnote{min\,mod($a,b$)=sign($a
$)min(abs($a$),abs($b$)) when $ab>0$, mod($a,b$)=0 when $ab\leq0$} of them,
then TVD property is preserved.

{\bf Theorem 1}. Consider the one dimensional hyperbolic conservation law
$$
	\frac{\p U}{\p t} + \frac{\p f(U)}{\p x} = 0	\eqno{(E)}
$$
and a $2k+1$ point finite difference scheme of it:
$$
	U^{n+1}_i = U^n_i -\Delta t \bar f(U^n_{k-1},...,U^n_{k+1})	\eqno{(S)}
$$
Suppose it is TVD, i. e., for any $\{U^n_i\}$,
$$
\sum_{i=-\infty}^\infty |U_{i+1}^{n+1}-U_i^{n+1}| \leq \sum_{i=-\infty}^\infty |U_{i+1}^{n}-U_i^{n}| 
								\eqno{(2)}
$$
and suppose for any $i$, $U^n_{i-1}=U^n_i=U^n_{i+1}$ implies $U^{n+1}_i=U^n_i$.
Then as along as $U^{n+1}_N,...,U^{n+1}_{N+k-1}$ satisfy
$$
|U^{n+1}_N-U^{n+1}_{N-1}|+...+|U^{n+1}_{N+k-1}-U^{n+1}_{N+k-2}|\leq|U^n_{N+k-1}-U^{n+1}_{N-1}|
								\eqno{(3)}
$$
we have the TVD property preserved, i. e.,
$$
\sum_{i=-\infty}^{N+k-2} |U_{i+1}^{n+1}-U_i^{n+1}| 
\leq \sum_{i=-\infty}^{N+k-2} |U_{i+1}^{n}-U_i^{n}| 
								\eqno{(4)}
$$
Inner point values $U^{n+1}_i (i\leq N-1)$ are calculated by the finite difference
scheme. Formulae (for $k$ points $U^{n+1}_N,...,U^{n+1}_{N+k-1}$ 
on/out of the boundary) satisfying (3) can be given refering to (1).


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