限制器
一些常用的流量限制器/斜率限制器
- CHARM [not 2nd order TVD] (Zhou, 1995)
\[\phi_{cm}(r)=\left\{ \begin{array}{ll}
\frac{r\left(3r+1\right)}{\left(r+1\right)^{2}}, \quad r>0, \quad\lim_{r\rightarrow\infty}\phi_{cm}(r)=3 \\
0 \quad \quad\, , \quad r\le 0
\end{array}\right.
\]
- HCUS [not 2nd order TVD] (Waterson & Deconinck, 1995)
\[\phi_{hc}(r) = \frac{ 1.5 \left(r+\left| r \right| \right)}{ \left(r+2 \right)} ; \quad \lim_{r \rightarrow \infty}\phi_{hc}(r) = 3
\]
- HQUICK [not 2nd order TVD] (Waterson & Deconinck, 1995)
\[\phi_{hq}(r) = \frac{2 \left(r + \left|r \right| \right)}{ \left(r+3 \right)} ; \quad \lim_{r \rightarrow \infty}\phi_{hq}(r) = 4
\]
- Koren (Koren, 1993) – third-order accurate for sufficiently smooth data[1]
\[\phi_{kn}(r) = \max \left[ 0, \min \left(2 r, \left(2 + r \right)/3, 2 \right) \right]; \quad \lim_{r \rightarrow \infty}\phi_{kn}(r) = 2
\]
- minmod – symmetric (Roe, 1986)
\[\phi_{mm} (r) = \max \left[ 0 , \min \left( 1 , r \right) \right] ; \quad \lim_{r \rightarrow \infty}\phi_{mm}(r) = 1
\]
- monotonized central (MC) – symmetric (van Leer, 1977)
\[\phi_{mc} (r) = \max \left[ 0 , \min \left( 2 r, 0.5 (1+r), 2 \right) \right] ; \quad \lim_{r \rightarrow \infty}\phi_{mc}(r) = 2
\]
- Osher (Chatkravathy and Osher, 1983)
\[\phi_{os} (r) = \max \left[ 0 , \min \left( r, \beta \right) \right], \quad \left(1 \leq \beta \leq 2 \right) ; \quad \lim_{r \rightarrow \infty}\phi_{os} (r) = \beta
\]
- ospre – symmetric (Waterson & Deconinck, 1995)
\[\phi_{op} (r) = \frac{1.5 \left(r^2 + r \right) }{\left(r^2 + r +1 \right)} ; \quad \lim_{r \rightarrow \infty}\phi_{op} (r) = 1.5
\]
- smart [not 2nd order TVD] (Gaskell & Lau, 1988)
\[\phi_{sm}(r) = \max \left[ 0, \min \left(2 r, \left(0.25 + 0.75 r \right), 4 \right) \right] ; \quad \lim_{r \rightarrow \infty}\phi_{sm}(r) = 4
\]
- superbee – symmetric (Roe, 1986)
\[\phi_{sb} (r) = \max \left[ 0, \min \left( 2 r , 1 \right), \min \left( r, 2 \right) \right] ; \quad \lim_{r \rightarrow \infty}\phi_{sb} (r) = 2
\]
- Sweby – symmetric (Sweby, 1984)
\[\phi_{sw} (r) = \max \left[ 0 , \min \left( \beta r, 1 \right), \min \left( r, \beta \right) \right], \quad \left(1 \leq \beta \leq 2 \right) ; \quad \lim_{r \rightarrow \infty}\phi_{sw} (r) = \beta
\]
- UMIST (Lien & Leschziner, 1994)
\[\phi_{um}(r) = \max \left[ 0, \min \left(2 r, \left(0.25 + 0.75 r \right), \left(0.75 + 0.25 r \right), 2 \right) \right] ; \quad \lim_{r \rightarrow \infty}\phi_{um}(r) = 2
\]
- van Albada 1 – symmetric (van Albada, et al., 1982)
\[\phi_{va1} (r) = \frac{r^2 + r}{r^2 + 1 } ; \quad \lim_{r \rightarrow \infty}\phi_{va1} (r) = 1
\]
- van Albada 2 – alternative form [not 2nd order TVD] used on high spatial order schemes (Kermani, 2003)
\[\phi_{va2} (r) = \frac{2 r}{r^2 + 1} ; \quad \lim_{r \rightarrow \infty}\phi_{va2} (r) = 0
\]
- van Leer – symmetric (van Leer, 1974)
\[\phi_{vl} (r) = \frac{r + \left| r \right| }{1 + \left| r \right| } ; \quad \lim_{r \rightarrow \infty}\phi_{vl} (r) = 2
\]
上面所有对称型限制器都具有如下对称性质：
\[\begin{eqnarray}
\begin{aligned}
\frac{ \phi \left( r \right)}{r} = \phi \left( \frac{1}{r} \right)
\end{aligned}
\end{eqnarray}
\]
这个对称性质可以保证限制过程不管是向前或者向后结果都是相同的。
除非明确指出，以上限制器函数都是二阶。这代表它们都设计为通过解的某个特殊区域，及TVD区域，来保证格式的稳定性。二阶精度，TVD限制器至少满足以下条件
-
\(r \le \phi(r) \le 2r, \left( 0 \le r \le 1 \right)\)
-
\(1 \le \phi(r) \le r, \left( 1 \le r \le 2 \right)\)
-
\(1 \le \phi(r) \le 2, \left( r > 2 \right)\)
-
\(\phi(1)=1\)
二阶TVD格式的允许区域如下图所示（Sweby Diagram），每个限制函数同时绘制在图中。在Osher和Sweby限制函数中，β取值为1.5
一般的minmod限制器
余下的一种限制器形式较特殊，val-Leer的单变量限制器(van Leer, 1979; Harten and Osher, 1987; Kurganov and Tadmor, 2000)。其形式如下:
\[\begin{equation}
\phi_{mg}(u,\theta)=\max\left(0,\min\left(\theta r,\frac{1+r}{2},\theta\right)\right),\quad\theta\in\left[1,2\right].
\end{equation}
\]
注意：当 θ=1时\(\phi_{mg}\)耗散形最强，当 θ=2 时， \(\phi_{mg}\)简化为 \(\phi_{mm}\)，耗散性最小。
