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