Commit fb3d288a by Julio

### 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.
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 ... ... @@ -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}
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