JUC: updated notes. I obtained the 4th order scheme. I now need to improve the...

JUC: updated notes. I obtained the 4th order scheme. I now need to improve the text and implement it.

notes/notes_on_fourth_order_FD.tex

@@ -107,14 +107,130 @@ and the following at the boundary:
     \left( \frac{\partial u}{\partial x} \right)_0 \approx \frac{1}{2}\left[ \left( \frac{\partial u}{\partial x} \right)_{\nicefrac{1}{2}} + \left( \frac{\partial u}{\partial x} \right)_{-\nicefrac{1}{2}} \right].
 \end{align}
 
-\subsection{Own approach}
+\subsection*{Own approach}
 
-We will use the following fourth order Compact Finite Differences approximation:
+First, we will use the chain rule
+
+\begin{align}
+    \frac{du}{dx}\Big|_{n} = \frac{du}{d\xi}\Big|_{n} \cdot \frac{d\xi}{dx}\Big|_{n} 
+\end{align}
+
+for the second term on the right hand side, we can use the mapping between grids:
+\begin{align}
+    x(\xi) = -c \ln(1-\xi) &\iff \xi(x) = 1 - e^{-\nicefrac{x}{c}} 
+\end{align}
+and calculate its derivative
+\begin{align}
+    \frac{\partial \xi}{\partial x}\Big|_n &= \frac{1}{c} e^{-\nicefrac{x}{c}}\Big|_n = \frac{1}{c}\left( 1 - \xi \right) = \frac{1}{c}\left( 1 - \frac{n}{N} \right).
+\end{align}
+We then have
+\begin{align}
+    \frac{du}{dx}\Big|_{n} = \frac{du}{d\xi}\Big|_{n} \cdot \frac{1}{c}\left( 1 - \frac{n}{N} \right)
+\end{align}
+
+We will approximate the remaining term by using the following fourth order Compact Finite Differences approximation, the so-called Pade's approximation:
 \begin{align*}
-    - 2 \frac{\partial u}{\partial \xi}\Big|_{n-\nicefrac{1}{2}} + \frac{\partial u}{\partial \xi}\Big|_{n} - 2 \frac{\partial u}{\partial \xi}\Big|_{n+\nicefrac{1}{2}} = -\frac{3}{4d} \left( u_{n+1} - u_{n-1} \right)
+    \frac{1}{4} \frac{\partial u}{\partial \xi}\Big|_{n-1} + \frac{\partial u}{\partial \xi}\Big|_{n} + \frac{1}{4}\frac{\partial u}{\partial \xi}\Big|_{n+1} = \frac{3}{4d} \left( u_{n+1} - u_{n-1} \right)
 \end{align*}
 
-obtained from the paper ``Compact Finite Difference Schemes with Spectral-like Resolution'' by Sanjiva K. Lele.
+with $d = \xi_{1} - \xi_{0}$, for the interior points, i.e. $0 < n < N-1$ and
+
+\begin{align*}
+    \frac{\partial u}{\partial \xi}\Big|_{0} + 3\frac{\partial u}{\partial \xi}\Big|_{1} = \frac{1}{d} \left( -\frac{17}{6}u_{1} + \frac{3}{2}u_{2} + \frac{3}{2}u_3 - \frac{1}{6}u_4 \right)
+\end{align*}
+
+for the left boundary. This scheme is obtained from the paper ``\emph{Compact Finite Difference Schemes with Spectral-like Resolution}'' by Sanjiva K. Lele, taking a fourth order approximation on both the boundary and the interior points.
+
+Therefore, we get the following system
+
+\begin{align*}
+    \underbrace{
+    \begin{bmatrix}
+        1               & 3               & 0               & 0              & 0                & \dots        & 0 \\
+        \nicefrac{1}{4} & 1               & \nicefrac{1}{4} & 0              & 0                & \ddots        & 0 \\
+        0               & \nicefrac{1}{4} & 1               & \nicefrac{1}{4}  & 0                & \ddots        & 0 \\
+        0               & 0               & \nicefrac{1}{4} & 1               & \nicefrac{1}{4}  & \ddots        & \vdots \\
+        \vdots          & \ddots          & \ddots & \ddots & \ddots  & \ddots        & \vdots \\
+        0 & \dots & \dots & \dots & 0 & \nicefrac{1}{4} & 1
+    \end{bmatrix}}_{=:M}
+    \underbrace{
+    \begin{bmatrix}
+        \frac{\partial u}{\partial \xi}\Big|_{0} \\
+        \frac{\partial u}{\partial \xi}\Big|_{1} \\
+        \frac{\partial u}{\partial \xi}\Big|_{2} \\
+        \vdots \\
+        \frac{\partial u}{\partial \xi}\Big|_{N-2}
+    \end{bmatrix}}_{=:\frac{\partial \bf{u}}{\partial \xi}}
+    +
+    \begin{bmatrix}
+        0 \\
+        0 \\
+        0 \\
+        \vdots \\
+        \cancelto{0}{\frac{\partial u}{\partial \xi}\Big|_{N-1}}
+    \end{bmatrix}
+    = \\
+    + \frac{1}{d}
+    \underbrace{
+    \begin{bmatrix}
+        -\nicefrac{17}{6} & \nicefrac{3}{2} & \nicefrac{3}{2} & -\nicefrac{1}{6} & 0 & \dots & 0 \\
+        -\nicefrac{3}{4} & 0 & \nicefrac{3}{4} & 0 & 0 & \ddots & 0 \\
+        0 & -\nicefrac{3}{4} & 0 & \nicefrac{3}{4} & 0 & \ddots & 0 \\
+        0 & 0 & -\nicefrac{3}{4} & 0 & \nicefrac{3}{4} & \ddots & \vdots \\
+        \vdots & \ddots & \ddots & \ddots & \ddots & \ddots & \vdots \\
+        0 & \dots & \dots & \dots & 0 & -\nicefrac{3}{4} & 0
+    \end{bmatrix}}_{=:A}
+    \underbrace{
+    \begin{bmatrix}
+        u_{0} \\
+        u_{1} \\
+        u_{2} \\
+        \vdots \\
+        u_{N-2}
+    \end{bmatrix}}_{=:\bf{u}}
+    +
+    \begin{bmatrix}
+        0 \\
+        0 \\
+        0 \\
+        \vdots \\
+        \cancelto{0}{u_{N-1}}
+    \end{bmatrix}
+\end{align*}
+
+in short
+
+\begin{align*}
+    M \frac{\partial \bf{u}}{\partial \xi} = \frac{1}{d} A \bf{u},
+\end{align*}
+therefore
+\begin{align*}
+    \frac{\partial \bf{u}}{\partial \xi} = \frac{1}{d} M^{-1} A \bf{u}.
+\end{align*}
+which can be used in the expression of the $x-$derivative
+\begin{align}
+    \frac{\partial \textbf{u}}{\partial x} = \mathcal{G} \frac{\partial\bf{u}}{\partial\xi} \approx \frac{1}{d} \mathcal{G} M^{-1} A \bf{u}.
+\end{align}
+
+where
+\begin{align}
+    \mathcal{G} :=
+    \begin{bmatrix}
+        \frac{\partial \xi}{\partial x}\Big|_0 & 0 & 0 & \dots & 0 \\
+        0 & \frac{\partial \xi}{\partial x}\Big|_1 & 0 & \dots & 0 \\
+        0 & 0 & \frac{\partial \xi}{\partial x}\Big|_2 & \dots & 0 \\
+        \vdots & \ddots & \ddots & \ddots & \vdots \\
+        0 & \dots & \dots & 0 & \frac{\partial \xi}{\partial x}\Big|_{N-1}
+    \end{bmatrix}
+\end{align}
+
+which is a first-derivative operator of order 4. We will then obtain a second order operator by considering this first-derivative operator twice,
+
+\begin{align}
+    \frac{\partial}{\partial x} \left( \frac{\partial \bf{u}}{\partial x} \right) &\approx \frac{1}{d} \mathcal{G} M^{-1} A \frac{\partial \bf{u}}{\partial x} \approx \\
+    &\approx\left(\frac{1}{d} \mathcal{G} M^{-1} A \right) \left( \frac{1}{d} \mathcal{G} M^{-1} A\right) \textbf{u} = \frac{1}{d^2} \left( \mathcal{G} M^{-1} A \right)^2 \bf{u}
+\end{align}
+
 
 
 \end{document}