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{ egin{array}{ll}
frac{rleft(3r+1ight)}{left(r+1ight)^{2}}, quad r>0, quadlim_{rightarrowinfty}phi_{cm}(r)=3 \
0 quad quad\, , quad rle 0
end{array}ight.
]

HCUS [not 2nd order TVD] (Waterson & Deconinck, 1995)

[phi_{hc}(r) = frac{ 1.5 left(r+left| r ight| ight)}{ left(r+2 ight)} ; quad lim_{r ightarrow infty}phi_{hc}(r) = 3
]

HQUICK [not 2nd order TVD] (Waterson & Deconinck, 1995)

[phi_{hq}(r) = frac{2 left(r + left|r ight| ight)}{ left(r+3 ight)} ; quad lim_{r ightarrow 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 ight)/3, 2 ight) ight]; quad lim_{r ightarrow infty}phi_{kn}(r) = 2
]

minmod – symmetric (Roe, 1986)

[phi_{mm} (r) = max left[ 0 , min left( 1 , r ight) ight] ; quad lim_{r ightarrow 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 ight) ight] ; quad lim_{r ightarrow infty}phi_{mc}(r) = 2
]

Osher (Chatkravathy and Osher, 1983)

[phi_{os} (r) = max left[ 0 , min left( r, eta ight) ight], quad left(1 leq eta leq 2 ight) ; quad lim_{r ightarrow infty}phi_{os} (r) = eta
]

ospre – symmetric (Waterson & Deconinck, 1995)

[phi_{op} (r) = frac{1.5 left(r^2 + r ight) }{left(r^2 + r +1 ight)} ; quad lim_{r ightarrow 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 ight), 4 ight) ight] ; quad lim_{r ightarrow infty}phi_{sm}(r) = 4
]

superbee – symmetric (Roe, 1986)

[phi_{sb} (r) = max left[ 0, min left( 2 r , 1 ight), min left( r, 2 ight) ight] ; quad lim_{r ightarrow infty}phi_{sb} (r) = 2
]

Sweby – symmetric (Sweby, 1984)

[phi_{sw} (r) = max left[ 0 , min left( eta r, 1 ight), min left( r, eta ight) ight], quad left(1 leq eta leq 2 ight) ; quad lim_{r ightarrow infty}phi_{sw} (r) = eta
]

UMIST (Lien & Leschziner, 1994)

[phi_{um}(r) = max left[ 0, min left(2 r, left(0.25 + 0.75 r ight), left(0.75 + 0.25 r ight), 2 ight) ight] ; quad lim_{r ightarrow 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 ightarrow 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 ightarrow infty}phi_{va2} (r) = 0
]

van Leer – symmetric (van Leer, 1974)

[phi_{vl} (r) = frac{r + left| r ight| }{1 + left| r ight| } ; quad lim_{r ightarrow infty}phi_{vl} (r) = 2
]

上面所有对称型限制器都具有如下对称性质：

[egin{eqnarray}
egin{aligned}
frac{ phi left( r ight)}{r} = phi left( frac{1}{r} ight)
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 ight))

(1 le phi(r) le r, left( 1 le r le 2 ight))

(1 le phi(r) le 2, left( r > 2 ight))

(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:

[egin{equation}
phi_{mg}(u, heta)=maxleft(0,minleft( heta r,frac{1+r}{2}, hetaight)ight),quad hetainleft[1,2ight].
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。