Wall Regularized Function¶
Wall Regular¶
The Wall is a function that is
\[w(-\infty) = 0,\qquad w(0)=\epsilon, \quad
w(h)=1,\qquad w(\infty)=\infty\]
and grows quadratically for \(x>0\). It is a linear combination of lines and hyperbolas:
\[\begin{split}w(x) := \begin{cases}
\dfrac{\epsilon}{1-z} & x < 0 \\[1em]
\epsilon +
a z \left(z-\dfrac{4}{3}\right)+
\dfrac{bz}{\frac{1}{4}+z} +
\dfrac{cz}{\frac{1}{2}+z} +
\dfrac{dz}{\frac{3}{4}+z} +
\dfrac{ez}{1+z}
& \textrm{otherwise},
\end{cases}\end{split}\]
where \(z:= x/h\) and the coefficients \(a\), \(b\), \(c\), \(d\) and \(e\) are defined as
\[a := \dfrac{315}{104} - \dfrac{1551}{208}\epsilon,\]
\[b := \dfrac{385}{3328} - \dfrac{10055}{19968}\epsilon,\]
\[c := \dfrac{243}{13}\epsilon - \dfrac{105}{26},\]
\[d := \dfrac{42525}{3328} - \dfrac{462105}{6656}\epsilon,\]
\[e := \dfrac{1882}{39}\epsilon - \dfrac{70}{13}\]
The result is shown in Figure XO:fig:wall. The first and second derivative are respectively:
\[\begin{split}hw'(x)=\begin{cases}
\dfrac{e}{(1+z)^2}+\dfrac{12d}{(3+4z)^2}+\dfrac{2c}{(1+2z)^2}+\dfrac{4b}{(1+4z)^2}+a & x \geq 0 \\[1em]
\dfrac{\epsilon}{(1-z)} & x < 0
\end{cases}\end{split}\]
\[\begin{split}h^2w''(x) = \begin{cases}
2a-\dfrac{2e}{(1+z)^3}-\dfrac{96d}{(3+4z)^3}-\dfrac{8c}{(1+2z)^3}-\dfrac{32b}{(1+4z)^3} & x \geq 0 \\[1em]
\dfrac{2\epsilon}{(1-z)^3}& x < 0
\end{cases}\end{split}\]
The coefficient are evaluated so that \(w(h)=1\) and \(w(0)=\epsilon\).
Wall Smooth¶
The Wall is a function that is
\[w(-\infty) = 0,\qquad w(0)=\epsilon, \qquad
w(h)=1,\qquad w(\infty)=\infty\]
and grows quadratically for \(x>0\).
\[w(x) = \dfrac{A}{2B^2}\left(\left(\frac{1}{2}+B^2x^2\right)(\mathrm{erf}(Bx)+1)+\dfrac{Bx}{\sqrt{\pi}}\exp(-B^2x^2)\right)\]
Wall Piecewise¶
The Wall is a function that is
\[w(x) = 0,\quad x\in(-\infty,0), \quad
w(h)=1,\qquad w(\infty)=\infty\]
and grows quadratically for \(x>0\).
\[\begin{split}w(x)=
\frac{1}{3}
\begin{cases}
0 & x<0 \\[1em]
z^4 \left( 5 - 2z\right) & x<h \\[1em]
3+10 z\left(z-1\right) & \textrm{otherwise}
\end{cases}
\qquad
z = \dfrac{x}{h}\end{split}\]