diff --git a/Project2/LyX/TheoryAndMethods.lyx b/Project2/LyX/TheoryAndMethods.lyx index 9f3a7ad6912f14f7683971e083ef3bf87cf3da41..4f569760354d0c743292e08e2ed380051c49d3d8 100644 --- a/Project2/LyX/TheoryAndMethods.lyx +++ b/Project2/LyX/TheoryAndMethods.lyx @@ -5,6 +5,9 @@ \save_transient_properties true \origin unavailable \textclass IEEEtran-CompSoc +\begin_preamble +\usepackage{tikz} +\end_preamble \use_default_options true \maintain_unincluded_children no \language american @@ -316,5 +319,764 @@ Therefore, it cannot be guaranteed that the PDE is fulfilled. \end_layout +\begin_layout Subsection +Formation of Discontinuities +\end_layout + +\begin_layout Standard +An advection equation may form discontinuities, + also known as shocks for +\begin_inset Formula $u_{x}\notin\mathcal{R}$ +\end_inset + +. + Therefore, + shocks are formed for +\begin_inset Formula $1+u_{0}'(x_{0})f''(u_{0}(x_{0}))t=0$ +\end_inset + + +\end_layout + +\begin_layout Proof +Shocks are formed for +\begin_inset Formula $1+u_{0}'(x_{0})f''(u_{0}(x_{0}))t=0$ +\end_inset + + +\end_layout + +\begin_layout Proof +Consider the solution for the general conservation law +\begin_inset Formula +\begin{align} +u(x,t) & =u_{0}\left(\underbrace{x-f'(u(x,t)t}_{v(x,t)}\right) +\end{align} + +\end_inset + +and calculate the spatial derivative ( +\begin_inset Formula $u_{x}$ +\end_inset + +) of this +\begin_inset Formula +\begin{align} +u_{x} & =\frac{\partial}{\partial x}u_{0}(v)=u_{0}'(v)\frac{\partial v}{\partial x}=u_{0}'(v)\frac{\partial}{\partial x}\left(x-f'(u)t\right)\\ + & =u_{0}'(v)\left(1-f''(u)u_{x}t\right)\\ +\Leftrightarrow u_{x}\left(1+u_{0}'(v)f''(u)t\right) & =u_{0}'(v)\\ +u_{x}(x,t) & =\frac{u_{0}'(v)}{1+u_{0}'(x)f''\left(u_{0}(x_{0})\right)t} +\end{align} + +\end_inset + +Conclude from this +\begin_inset Formula +\begin{align*} +u(x,t)\text{ is continuous } & \forall u_{x}\in\mathcal{R}\\ +u(x,t)\text{ is discontinuous if } & 1+u_{0}'(x)f''\left(u_{0}(x_{0})\right)t=0 +\end{align*} + +\end_inset + + +\end_layout + +\begin_layout Subsubsection +Formation of Discontinuities for the case +\begin_inset Formula $f(u)=u^{2}$ +\end_inset + + +\end_layout + +\begin_layout Standard +Consider the convex, + scalar function +\begin_inset Formula $f(u)=u^{2}$ +\end_inset + + with the derivatives +\begin_inset Formula +\[ +f'(u)=2u\quad f''(u)=2 +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +And the initial condition +\begin_inset Formula +\begin{equation} +u_{0}(x)=\begin{cases} +2x & ,0\leq x\leq\frac{1}{2}\\ +2(1-x) & ,\frac{1}{2}<x\leq1 +\end{cases}\qquad u_{0}'(x)=\begin{cases} +2 & ,0\leq x\leq\frac{1}{2}\\ +-2 & ,\frac{1}{2}<x\leq1 +\end{cases} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Consider the right part of the domain ( +\begin_inset Formula $\frac{1}{2}<x\leq1$ +\end_inset + +) and insert +\begin_inset Formula $u_{0}'(x_{0})$ +\end_inset + + and +\begin_inset Formula $f''(u_{0}(x_{0}))$ +\end_inset + + into the discontinuity condition +\begin_inset Formula +\begin{align} +1+u_{0}'(x)f''\left(u_{0}(x_{0})\right)t & =0\\ +1+(-2)2t & =0\\ +1-4t & =0\\ +\Leftrightarrow t & =\frac{1}{4} +\end{align} + +\end_inset + +Conclude from this that for time +\begin_inset Formula $t=\frac{1}{4}$ +\end_inset + + a first shock forms. +\end_layout + +\begin_layout Subsection +Rarefaction waves +\end_layout + +\begin_layout Standard +\begin_inset Float figure +placement document +alignment document +wide false +sideways false +status collapsed + +\begin_layout Plain Layout +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +resizebox{ +\backslash +textwidth}{!}{ +\end_layout + +\begin_layout Plain Layout + + +\backslash +begin{tikzpicture} +\end_layout + +\begin_layout Plain Layout + + % Axes +\end_layout + +\begin_layout Plain Layout + + +\backslash +draw[->] (0,0) -- (1.5,0) node[right] {$x$}; + % x-axis +\end_layout + +\begin_layout Plain Layout + + +\backslash +draw[->] (0,0) -- (0,1.5) node[above] {$t$}; + % t-axis +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + + % Rarefaction Wave Lines +\end_layout + +\begin_layout Plain Layout + + % Line for x_0 = 0 +\end_layout + +\begin_layout Plain Layout + + +\backslash +draw[thick] (0, + 0) -- (0, + 1); + % Vertical line at x=0 +\end_layout + +\begin_layout Plain Layout + + +\backslash +node at (0,-0.3) {$x_0=0$}; +\end_layout + +\begin_layout Plain Layout + + +\end_layout + +\begin_layout Plain Layout + + % Line for x_0 = 1/4: + y=1/4+1/2*t +\end_layout + +\begin_layout Plain Layout + + +\backslash +draw[thick] (1/4,0) -- (1/3,1); +\end_layout + +\begin_layout Plain Layout + + +\backslash +node at (1/4,-0.3) {$x_0= +\backslash +frac{1}{4}$}; +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + + % Line for x_0 = 1/2: + y=1/2 + t +\end_layout + +\begin_layout Plain Layout + + +\backslash +draw[thick] (1/2,0) -- (3/4,1); +\end_layout + +\begin_layout Plain Layout + + +\backslash +node at (1/2,-0.3) {$x_0= +\backslash +frac{1}{2}$}; +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + + % Line for x_0 = 3/4: + x=3/4 + 1/2 t +\end_layout + +\begin_layout Plain Layout + + +\backslash +draw[thick] (3/4,0) -- (9/10,1); +\end_layout + +\begin_layout Plain Layout + + +\backslash +node at (3/4,-0.3) {$x_0= +\backslash +frac{3}{4}$}; +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + + % Line for x_0 = 1: + x=1 +\end_layout + +\begin_layout Plain Layout + + +\backslash +draw[thick] (1,0) -- (1,1); +\end_layout + +\begin_layout Plain Layout + + +\backslash +node at (1,-0.3) {$x_0=1$}; +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + + +\backslash +end{tikzpicture} +\end_layout + +\begin_layout Plain Layout + +} +\end_layout + +\end_inset + + +\begin_inset Caption Standard + +\begin_layout Plain Layout +Rarefaction and Compression wave +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Subsection +Rankine-Hugenoit Condition +\end_layout + +\begin_layout Standard +Consider the PDE +\begin_inset Formula $u_{t}+f(u)_{x}=0$ +\end_inset + + with the known solution +\begin_inset Formula $u(x,t)$ +\end_inset + +. + The Rankine-Hugenoit condition yields an, + explicit, + expression for the speed of a shock ( +\begin_inset Formula $s$ +\end_inset + +), + given a shock occurs: +\begin_inset Formula +\begin{equation} +s=\frac{f(u_{l})-f(u_{r})}{u_{l}-u_{r}} +\end{equation} + +\end_inset + +with +\begin_inset Formula $u_{l}$ +\end_inset + + the (constant) value on the left of the shock and +\begin_inset Formula $u_{r}$ +\end_inset + + the value on the right. + +\end_layout + +\begin_layout Proof +The Rankine-Hugenoit Conditon +\begin_inset Formula $s=\frac{f(u_{l})-f(u_{r})}{u_{l}-u_{r}}$ +\end_inset + + +\end_layout + +\begin_layout Proof +Consider the following assumptions: +\end_layout + +\begin_layout Proof +The solution to the scalar conservation problem +\begin_inset Formula $u_{t}+f(u)_{x}$ +\end_inset + + is yields a shock +\end_layout + +\begin_layout Proof +The shock, + moving with a positive velocity, + along a path is represented by +\begin_inset Formula $u_{l}(t)$ +\end_inset + + and +\begin_inset Formula $u_{r}(t)$ +\end_inset + + +\end_layout + +\begin_layout Proof +\begin_inset Formula $u_{l}$ +\end_inset + + and +\begin_inset Formula $u_{r}$ +\end_inset + + are locally constant +\end_layout + +\begin_layout Proof +The shock speed is represented by +\begin_inset Formula $s=\frac{\Delta x}{\Delta t}$ +\end_inset + + +\end_layout + +\begin_layout Proof +Consider the window +\begin_inset Formula $R=[x_{1},x_{1}+\Delta x]\times[t_{1},t_{1}+\Delta t]$ +\end_inset + + around the shock. +\end_layout + +\begin_layout Proof +\begin_inset Float figure +placement document +alignment document +wide false +sideways false +status open + +\begin_layout Plain Layout +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +begin{tikzpicture} +\end_layout + +\begin_layout Plain Layout + + % Axes +\end_layout + +\begin_layout Plain Layout + + +\backslash +draw[->] (0,0) -- (4,0) node[right] {$x$}; + % x-axis +\end_layout + +\begin_layout Plain Layout + + +\backslash +draw[->] (0,0) -- (0,3.5) node[above] {$t$}; + % t-axis +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + + % Window R indication +\end_layout + +\begin_layout Plain Layout + + +\backslash +fill[gray!20] (1,1) rectangle(3,2); + +\end_layout + +\begin_layout Plain Layout + + +\backslash +node at(2,2.25) {R}; +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + + % Function u(x,t) +\end_layout + +\begin_layout Plain Layout + + +\backslash +draw[thick, + domain=1:3] plot ( +\backslash +x,{1.0 + 0.5*(( +\backslash +x))^3/3^3*(2}) node[right] {}; + % Example parabola function +\end_layout + +\begin_layout Plain Layout + + +\backslash +node at (3.6,2.25) {$u(x,t)$}; +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + + % Points and intervals +\end_layout + +\begin_layout Plain Layout + + +\backslash +draw[dashed] (1,0) -- (1,3.5); +\end_layout + +\begin_layout Plain Layout + + +\backslash +draw[dashed] (3,0) -- (3,3.5); +\end_layout + +\begin_layout Plain Layout + + +\end_layout + +\begin_layout Plain Layout + + +\backslash +node at (1,-0.3) {$x_1$}; +\end_layout + +\begin_layout Plain Layout + + +\backslash +node at (3,-0.3) {$x_1 + +\backslash +Delta x$}; +\end_layout + +\begin_layout Plain Layout + + +\end_layout + +\begin_layout Plain Layout + + % Time intervals +\end_layout + +\begin_layout Plain Layout + + +\backslash +draw[dashed] (0,1) -- (4,1); +\end_layout + +\begin_layout Plain Layout + + +\backslash +draw[dashed] (0,2) -- (4,2); +\end_layout + +\begin_layout Plain Layout + + +\end_layout + +\begin_layout Plain Layout + + +\backslash +node at (-0.3, + 1) {$t_1$}; +\end_layout + +\begin_layout Plain Layout + + +\backslash +node at (-0.7, + 2) {$t_1 + +\backslash +Delta t$}; +\end_layout + +\begin_layout Plain Layout + + +\end_layout + +\begin_layout Plain Layout + + % Indicating u_l and u_r +\end_layout + +\begin_layout Plain Layout + + +\backslash +node at (2.0,1.5) {$u_l$}; +\end_layout + +\begin_layout Plain Layout + + +\backslash +node at (2.5,1.2) {$u_r$}; +\end_layout + +\begin_layout Plain Layout + + +\backslash +end{tikzpicture} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +Schematic Figure of Rankine-Hugenoit condition window +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Proof +Start by integrating the PDE around the shock (the window +\begin_inset Formula $R$ +\end_inset + +) over space and time +\begin_inset Formula +\begin{align} +0 & =\iint_{R}\left(u_{t}+f(u)_{x}\right)dxdt\\ + & =\int_{t_{1}}^{t_{1}+\Delta t}\frac{d}{dt}\int_{x_{1}}^{x_{1}+\Delta x}udxdt+\int_{t_{1}}^{t_{1}+\Delta t}f(u)\bigg|_{x_{1}}^{x_{1}+\Delta x}dt\\ + & =\int_{x_{1}}^{x_{1}+\Delta x}u\bigg|_{t_{1}}^{t_{1}+\Delta t}dx+\int_{t_{1}}^{t_{1}+\Delta t}f(u)\bigg|_{x_{1}}^{x_{1}+\Delta x}dt +\end{align} + +\end_inset + +Now, + consider the assumption that the scalars +\begin_inset Formula $u_{l}$ +\end_inset + + and +\begin_inset Formula $u_{r}$ +\end_inset + + are constant in the window +\begin_inset Formula $R$ +\end_inset + + and that +\begin_inset Formula $R$ +\end_inset + + is small. + With this evaluate +\begin_inset Formula +\begin{equation} +\begin{array}{cc} +u(t_{1})=u_{r} & u(t_{1}+\Delta t)=u_{l}\\ +f(u(x_{1}))=f(u_{r}) & f(u(x_{1}+\Delta x))=u_{r} +\end{array} +\end{equation} + +\end_inset + +and simplify the integrals to +\begin_inset Formula +\begin{align} +0 & =\int_{x_{1}}^{x_{1}+\Delta x}\underbrace{(u_{l}-u_{r})}_{\text{constant}}dx+\int_{t_{1}}^{t_{1}+\Delta t}\left(f(u_{r})-f(u_{l})\right)dt\\ + & =(u_{l}-u_{r})\Delta x+\left(f(u_{r})-f(u_{l})\right)\Delta t\\ +\Leftrightarrow\frac{\Delta x}{\Delta t} & =\frac{f(u_{l})-f(u_{r})}{u_{l}-u_{r}}=s +\end{align} + +\end_inset + + +\end_layout + \end_body \end_document