diff --git a/Project2/LyX/TheoryAndMethods.lyx b/Project2/LyX/TheoryAndMethods.lyx
index 4f569760354d0c743292e08e2ed380051c49d3d8..6061c913ddcb2f09f2113b1ea65218ceef986a78 100644
--- a/Project2/LyX/TheoryAndMethods.lyx
+++ b/Project2/LyX/TheoryAndMethods.lyx
@@ -683,7 +683,7 @@ Rarefaction and Compression wave
 \end_layout
 
 \begin_layout Subsection
-Rankine-Hugenoit Condition
+Rankine-Hugoniot Condition
 \end_layout
 
 \begin_layout Standard
@@ -696,7 +696,7 @@ Consider the PDE
 \end_inset
 
 .
- The Rankine-Hugenoit condition yields an,
+ The Rankine-Hugoniot condition yields an,
  explicit,
  expression for the speed of a shock (
 \begin_inset Formula $s$
@@ -724,7 +724,7 @@ with
 \end_layout
 
 \begin_layout Proof
-The Rankine-Hugenoit Conditon 
+The Rankine-Hugoniot Conditon 
 \begin_inset Formula $s=\frac{f(u_{l})-f(u_{r})}{u_{l}-u_{r}}$
 \end_inset
 
@@ -1008,7 +1008,7 @@ end{tikzpicture}
 \begin_inset Caption Standard
 
 \begin_layout Plain Layout
-Schematic Figure of Rankine-Hugenoit condition window
+Schematic Figure of Rankine-Hugoniot condition window
 \end_layout
 
 \end_inset
@@ -1070,7 +1070,7 @@ and simplify the integrals to
 \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
+\Leftrightarrow\frac{\Delta x}{\Delta t} & =\frac{f(u_{l})-f(u_{r})}{u_{l}-u_{r}}=s\label{eq:RH}
 \end{align}
 
 \end_inset
@@ -1078,5 +1078,455 @@ and simplify the integrals to
 
 \end_layout
 
+\begin_layout Subsection
+The Class of Riemann Problems
+\end_layout
+
+\begin_layout Standard
+A Riemann problem is a specific problem for the scalar conservation law,
+ inspected above,
+ which reads
+\begin_inset Formula 
+\begin{equation}
+\begin{cases}
+u_{t}+f(u)_{x}=0\\
+u(x,t=0)=u_{0}(x) & =\begin{cases}
+u_{l} & ,x<0\\
+u_{r} & ,x>0
+\end{cases}
+\end{cases}
+\end{equation}
+
+\end_inset
+
+
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+Add the sketch plot
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+For the Riemann problem there can be found two different classes of solutions
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Similarity Solution
+\series default
+:
+ A similarity solution connects the two initial states by a continuous solution
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Shock Solution:
+
+\series default
+ A shock solution connects 
+\begin_inset Formula $u_{l}$
+\end_inset
+
+ and 
+\begin_inset Formula $u_{r}$
+\end_inset
+
+ by a discontinuous solution,
+ where the discontinuity moves with the speed s,
+ derived by the Rankine-Hugoniot condition
+\end_layout
+
+\begin_layout Standard
+In the following assume the flux function to be convex.
+ All derivates can be performed for a concave flux function in the same manner,
+ but will produce different results.
+\end_layout
+
+\begin_layout Subsubsection
+Similarity Solution
+\end_layout
+
+\begin_layout Standard
+A continuous similarity solution occurs under the condition 
+\begin_inset Formula $f'(u_{l})<f'(u_{r})$
+\end_inset
+
+.
+ For the Riemann problem the solution is undefined for 
+\begin_inset Formula $x=0$
+\end_inset
+
+,
+ where two different characteristics,
+ which will not cross.
+\begin_inset Formula 
+\[
+x=f'(u_{l})t\qquad x=f'(u_{r})t
+\]
+
+\end_inset
+
+In between these two characteristics,
+ there is one more valid characteristic,
+ namely 
+\begin_inset Formula $x=kt$
+\end_inset
+
+ with 
+\begin_inset Formula $f'(u_{l})<k<f'(u_{r})$
+\end_inset
+
+ for 
+\begin_inset Formula $k=\frac{x}{t}$
+\end_inset
+
+.
+ The scalar solution is of the form
+\begin_inset Formula 
+\begin{equation}
+u(x,t)=\begin{cases}
+u_{l} & ,\frac{x}{t}\leq f'(u_{l})\\
+v(\frac{x}{t}) & ,f'(u_{l}<x<f'(u_{r})\\
+u_{r} & ,\frac{x}{t}\geq f'(u_{r})
+\end{cases}\label{eq:SimilaritySolutionAbstract}
+\end{equation}
+
+\end_inset
+
+The arising question is how to calculate 
+\begin_inset Formula $v(\frac{x}{t})=u(x,t)$
+\end_inset
+
+.
+ For this,
+ insert the partial derivatives
+\begin_inset Formula 
+\begin{align}
+u_{t}(x,t) & =\frac{\partial}{\partial t}v\left(\frac{x}{t}\right)=v'\left(\frac{x}{t}\right)\frac{\partial}{\partial t}\left(\frac{x}{t}\right)=\frac{-x}{t^{2}}v'\left(\frac{x}{t}\right)\\
+u_{x}(x,t) & =\frac{\partial}{\partial x}v\left(\frac{x}{t}\right)=v'(\left(\frac{x}{t}\right)\frac{\partial}{\partial x}\left(\frac{x}{t}\right)=\frac{1}{t}v'\left(\frac{x}{t}\right)\\
+f(u)_{x} & =f'(u)u_{x}=f'\left(v\left(\frac{x}{t}\right)\right)v'\left(\frac{x}{t}\right)\frac{1}{t}
+\end{align}
+
+\end_inset
+
+ into the partial differential equation
+\begin_inset Formula 
+\begin{align}
+0 & =u_{t}+f(u)_{x}\\
+ & =\frac{-x}{t^{2}}v'\left(\frac{x}{t}\right)+f'\left(v\left(\frac{x}{t}\right)\right)v'\left(\frac{x}{t}\right)\frac{1}{t}\quad\bigg|\cdot t\\
+ & =v'\left(\frac{x}{t}\right)\left[\frac{-x}{t}+f'\left(v\left(\frac{x}{t}\right)\right)\right]
+\end{align}
+
+\end_inset
+
+This equation can be fulfilled in two ways,
+ either let 
+\begin_inset Formula $v'\left(\frac{x}{t}\right)=0$
+\end_inset
+
+.
+ However,
+ this would imply 
+\begin_inset Formula $v\left(\frac{x}{t}\right)=u(x,t)$
+\end_inset
+
+ to be constant and can only be valid if 
+\begin_inset Formula $u_{l}=u_{r}$
+\end_inset
+
+.
+ The second possibility to fulfill the equation is to ask for
+\begin_inset Formula 
+\begin{align}
+\frac{-x}{t}+f'\left(v\left(\frac{x}{t}\right)\right) & =0\\
+\Leftrightarrow f'\left(v\left(\frac{x}{t}\right)\right) & =\frac{x}{t}\\
+\Leftrightarrow\left(f'\right)^{-1}\left(\frac{x}{t}\right) & =\left(f'\right)^{-1}\left(f\left(v\left(\frac{x}{t}\right)\right)\right)=v\left(\frac{x}{t}\right)\label{eq:v_x/t}
+\end{align}
+
+\end_inset
+
+Where 
+\begin_inset Formula $\left(f'\right)^{-1}$
+\end_inset
+
+ is the inverse of the derivative of the flux function.
+ Note that this inverse exists,
+ as the flux function is convex and therefore the second derivative is positive.
+ Now,
+ insert the solution 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:v_x/t"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+ into the general solution 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:SimilaritySolutionAbstract"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+ to get the solution
+\begin_inset Formula 
+\begin{equation}
+u(x,t)=\begin{cases}
+u_{l} & ,\frac{x}{t}\leq f'(u_{l})\\
+\left(f'\right)^{-1}\left(\frac{x}{t}\right) & ,f'(u_{l})<\frac{x}{t}<f'(u_{r})\\
+u_{r} & ,\frac{x}{t}\geq f'(u_{r})
+\end{cases}
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsubsection
+Shock Solution
+\end_layout
+
+\begin_layout Standard
+Contrary to the similarity solution,
+ the shock solution yields a discontinuous solution to the Riemann problem under the condition 
+\begin_inset Formula $f'(u_{l})>f'(u_{r})$
+\end_inset
+
+.
+ For the shock solution two characteristics cross,
+ yielding a discontinuous solution with a jump.
+ This jump moves with the shock speed 
+\begin_inset Formula $s$
+\end_inset
+
+,
+ derived by the Rankine-Hugoniot condition (equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:RH"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+).
+ The final solution reads
+\begin_inset Formula 
+\begin{equation}
+u(x,t)=\begin{cases}
+u_{l} & ,\frac{x}{t}\leq s\\
+u_{r} & ,x<\frac{x}{t}
+\end{cases}\qquad\text{with: }s=\frac{f(u_{l})-f(u_{r})}{u_{l}-u_{r}}
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsubsection
+Weak Solutions
+\end_layout
+
+\begin_layout Standard
+Weak solutions can simplify mathematical problems by introducing some test function 
+\begin_inset Formula $\Phi$
+\end_inset
+
+.
+ The original problem will be multiplied by the test function and integrated over the domain 
+\begin_inset Formula $\Omega$
+\end_inset
+
+.
+ Weak solutions have two important properties
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+No Derivatives:
+
+\series default
+ Weak solutions involve no derivatives in 
+\begin_inset Formula $u$
+\end_inset
+
+ and 
+\begin_inset Formula $f(u)$
+\end_inset
+
+ as they are moved to the test function
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Solution Space:
+
+\series default
+ The solution space of weak solutions is much larger then the solution space for strong solutions.
+ Weak solutions can even include more then one solution for a given problem
+\end_layout
+
+\begin_layout Standard
+For a weak solution consider a smooth test function 
+\begin_inset Formula $\Phi(x,t)$
+\end_inset
+
+ with compact support (the test function is zero outside some finite box).
+ To derive the weak form of the problem multiply the PDE with the test function and integrate over the domain 
+\begin_inset Formula $\Omega=\underbrace{[x_{1},x_{2}]}_{\mathcal{R}}\times\underbrace{[t_{1},t_{2}]}_{\mathcal{R}^{+}}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+The weak form then reads:
+ Find 
+\begin_inset Formula $u$
+\end_inset
+
+ s.t.
+ 
+\begin_inset Formula $\forall\Phi\in C_{0}^{1}(\mathcal{R}\times\mathcal{R}^{+})$
+\end_inset
+
+:
+\begin_inset Formula 
+\begin{align}
+0 & =\iint_{\Omega}\left[u_{t}+f(u)_{x}\right]\Phi(x,t)d\Omega\\
+ & =\int_{0}^{\infty}\int_{-\infty}^{\infty}\left(u_{t}\Phi(x,t)+f(u)_{x}\Phi(x,t)\right)dxdt\\
+ & =\int_{0}^{\infty}\int_{-\infty}^{\infty}\left(u\Phi_{t}(x,t)+f(u)\Phi_{x}(x,t)\right)dxdt+\int_{-\infty}^{\infty}u_{0}(x)\Phi(x,t=0)dx\label{eq:WeakFormAbstract}
+\end{align}
+
+\end_inset
+
+Note that in the last step integration by parts and the property that the test function vanishes on the boundaries was utilized.
+ Any function 
+\begin_inset Formula $u(x,t)$
+\end_inset
+
+ that fulfills the weak form (equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:WeakFormAbstract"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+) is a weak solution to the initial given problem.
+ This yields another problem,
+ as the solution 
+\begin_inset Formula $u$
+\end_inset
+
+ is not necessarily unique.
+ To determine if a weak solution is the correct solution to a specific problem,
+ the (Lax) entropy condition is introduced.
+\end_layout
+
+\begin_layout Definition
+Lax Entropy Condition
+\end_layout
+
+\begin_layout Definition
+A weak shock solution is a strong solution for a problem,
+ if,
+ and only if,
+ the shock satisfies 
+\begin_inset Formula 
+\begin{equation}
+f'(u_{l})>s>f'(u_{r})
+\end{equation}
+
+\end_inset
+
+ with the shock speed 
+\begin_inset Formula $s$
+\end_inset
+
+.
+ Conclude from this
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Jump Solution:
+
+\series default
+ A jump solution is a valid solution if 
+\begin_inset Formula $u_{l}>u_{r}$
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+The flux function is convex and yields 
+\begin_inset Formula $f'(u_{l})>f'(u_{r})$
+\end_inset
+
+.
+ Conclude 
+\begin_inset Formula $f'(u_{l})>s>f'(u_{r})$
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+
+\series bold
+Similarity Solution:
+
+\series default
+ The similarity solution is a valid solution if 
+\begin_inset Formula $u_{l}<u_{r}$
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+The flux function is convex and yields 
+\begin_inset Formula $f'(u_{l})<f'(u_{r})$
+\end_inset
+
+.
+ Conclude that no discontinuity of the form 
+\begin_inset Formula $f'(u_{l})>s>f'(u_{r})$
+\end_inset
+
+ can occur.
+\end_layout
+
+\end_deeper
 \end_body
 \end_document
diff --git a/Project2/LyX/main.lyx b/Project2/LyX/main.lyx
index eeb67936037136a6363e7b4d5d90157f104335f0..e7bdad92438b7555cb6a762664a01e3724753931 100644
--- a/Project2/LyX/main.lyx
+++ b/Project2/LyX/main.lyx
@@ -5,6 +5,9 @@
 \save_transient_properties true
 \origin unavailable
 \textclass IEEEtran-CompSoc
+\begin_preamble
+\usepackage{tikz}
+\end_preamble
 \use_default_options true
 \begin_modules
 eqs-within-sections
@@ -152,7 +155,7 @@ figs-within-sections
 \color background background
 \end_branch
 \branch List of Variables
-\selected 1
+\selected 0
 \filename_suffix 0
 \color background background
 \end_branch