Add Introduction

Project2/LyX/Introduction.lyx

@@ -100,6 +100,442 @@ name "sec:Introduction"
 \end_layout
 
 \begin_layout Standard
+Traffic jams occur in our everyday life.
+ Most people use the road,
+ either by car,
+ bus or bike,
+ every day to get to their job,
+ do grocery shopping,
+ to meet friends or to get to their hobbies.
+ Through this enormous use of the road,
+ traffic jams occur.
+ Traffic jams lead to a smaller speed of vehicles,
+ if too many vehicles are close to each other.
+\end_layout
+
+\begin_layout Standard
+The arising engineering question is,
+ how to reduce these traffic jams?
+ With a reduction of traffic jams,
+ the traffic quality can increase,
+ with less time on the road.
+ However,
+ to reduce traffic jams it is essential to understand how these work.
+ 
+\end_layout
+
+\begin_layout Standard
+For this,
+ it is necessary to study the underlying mathematical model of traffic jams,
+ which states a smaller vehicle speed at higher number densities.
+\begin_inset Float figure
+placement document
+alignment document
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\begin_inset Graphics
+	filename Figures/TrafficJam.jpg
+	lyxscale 5
+	width 60text%
+
+\end_inset
+
+
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+Image by Al Gг from Pixabay
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+Engineering Problem:
+\end_layout
+
+\begin_layout Itemize
+Traffic jams occur in our everyday life
+\end_layout
+
+\begin_deeper
+\begin_layout Itemize
+Many people use the road on an every day basis
+\end_layout
+
+\begin_deeper
+\begin_layout Itemize
+Get to job,
+ grocery shopping,
+ meeting friends,
+ get to sports
+\end_layout
+
+\begin_layout Itemize
+Car,
+ bus,
+ bike
+\end_layout
+
+\end_deeper
+\begin_layout Itemize
+We want to reduce these traffic jams,
+ but how?
+\end_layout
+
+\begin_layout Itemize
+How do we need to construct good roads,
+ speed-limits to reduce traffic jam to a minimum?
+\end_layout
+
+\begin_deeper
+\begin_layout Itemize
+For these need to understand the underlying mathematical problem of a traffic jam
+\end_layout
+
+\begin_layout Itemize
+smaller velocity at higher density
+\end_layout
+
+\end_deeper
+\begin_layout Itemize
+\begin_inset Float figure
+placement document
+alignment document
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\begin_inset Graphics
+	filename Figures/TrafficJam.jpg
+	lyxscale 5
+	width 60text%
+
+\end_inset
+
+
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+Image by Al Gг from Pixabay
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+The Mathematical Model
+\end_layout
+
+\begin_layout Standard
+The underlying mathematical model of traffic jams relies on the general conservation of mass
+\begin_inset Formula 
+\begin{equation}
+u_{t}+\left(V(u)u\right)_{x}=0,\quad x\in[a,b]
+\end{equation}
+
+\end_inset
+
+where 
+\begin_inset Formula $u$
+\end_inset
+
+ is the number density of vehicles (
+\begin_inset Formula $u\in[a,b]$
+\end_inset
+
+),
+ 
+\begin_inset Formula $V(u)$
+\end_inset
+
+ is the velocity of vehicles,
+ depending on the number density and 
+\begin_inset Formula $b-a$
+\end_inset
+
+ is the length of the road.
+ Furthermore,
+ an inflow boundary condition
+\begin_inset Formula 
+\begin{equation}
+u\big|_{x=a}=u_{\text{in}}
+\end{equation}
+
+\end_inset
+
+is assumed to hold true.
+ This inflow boundary condition models the number of arriving cars at the beginning of the road.
+\end_layout
+
+\begin_layout Standard
+Finally,
+ the problem can be reformulated to the general,
+ scalar,
+ nonlinear conservation law
+\begin_inset Formula 
+\begin{align}
+u_{t}+f(u)_{x} & =0\quad\text{in}x\in\Omega\\
+u\big|_{x=x_{\text{in}}} & =u_{\text{in}}
+\end{align}
+
+\end_inset
+
+
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+Mathematical model
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $u_{t}+\left(V(u)u\right)_{x}=0,\quad x\in[a,b]$
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
+\begin_layout Itemize
+with 
+\begin_inset Formula $b-a$
+\end_inset
+
+ length of road
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $V(u)$
+\end_inset
+
+ velocity of vehicles
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $u=u(x,t)\in[0,1]$
+\end_inset
+
+ number density of vehicles
+\end_layout
+
+\begin_layout Itemize
+Inflow boundary condition:
+ 
+\begin_inset Formula $u\bigg|_{x=a}=u_{\text{in}}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+Get the general,
+ scalar nonlinear conservation law:
+ 
+\begin_inset Formula $u_{t}+f(u)_{x}=0$
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+The main assumption for this model is,
+ that the vehicle velocity depends on the number density 
+\begin_inset Formula 
+\begin{equation}
+V(u)\propto(1-u)
+\end{equation}
+
+\end_inset
+
+Increasing number densities lead to a decreased vehicle velocity and vice versa.
+ Finally,
+ the velocity is scaled to its maximum value 
+\begin_inset Formula $V_{\text{max}}$
+\end_inset
+
+ and this maximum value is set to 
+\begin_inset Formula $1$
+\end_inset
+
+ (
+\begin_inset Formula $V_{\text{max}}=1$
+\end_inset
+
+),
+ resulting in
+\begin_inset Formula 
+\begin{align}
+V(u) & =V_{\text{max}}(1-u)\\
+ & =(1-u)\\
+\Rightarrow u_{t}+\left(u(1-u)\right)_{x} & =0,\quad x\in[a,b]\\
+\Leftrightarrow u_{t}+\left(f(u)\right)_{x} & =0\\
+\text{for: }f(u) & =u(1-u)
+\end{align}
+
+\end_inset
+
+ 
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+Main assumption in the construction of model is mass balance 
+\begin_inset Formula $u_{t}+(V(u)u)_{x}=0$
+\end_inset
+
+ with fluid velocity 
+\begin_inset Formula $V(u)=V_{\text{max}}(1-u)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+What is the underlying assumption of 
+\begin_inset Formula $V(u)$
+\end_inset
+
+?
+\end_layout
+
+\begin_deeper
+\begin_layout Itemize
+The more cars (
+\begin_inset Formula $u$
+\end_inset
+
+ increases),
+ the lower the speed 
+\begin_inset Formula $V(u)$
+\end_inset
+
+ decreases
+\end_layout
+
+\begin_layout Itemize
+Use 
+\begin_inset Formula $V(u)=V_{\text{max}}(1-u)$
+\end_inset
+
+ with 
+\begin_inset Formula $V_{\text{max}}$
+\end_inset
+
+ as constant maximum velocity (
+\begin_inset Formula $V_{\text{max}}=1$
+\end_inset
+
+),
+ arrive at
+\end_layout
+
+\begin_deeper
+\begin_layout Itemize
+\begin_inset Formula $u_{t}+\left(u(1-u)\right)_{x}=0,\quad x\in[a,b]$
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\end_deeper
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Arising Research Questions
+\end_layout
+
+\begin_layout Standard
+From this mathematical model and the physical background some research questions arise.
+\end_layout
+
+\begin_layout Enumerate
+How does the number density evolve over time?
+ What influence has the initial distribution on the transient behavior of the number densities?
+\end_layout
+
+\begin_layout Enumerate
+What is the highest number density?
+ Is there a mathematical explanation for this highest number density?
+\end_layout
+
+\begin_layout Enumerate
+What is the flux of the moving vehicles?
+ How does it change over time?
+ When is it high and when is it low?
+ 
+\end_layout
+
+\begin_layout Enumerate
+Is it more efficient to have a high distance or short distance between the cars?
+ With a higher distance a higher speed is possible,
+ but less cars are on the road.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+Relevant research questions
+\end_layout
+
+\begin_layout Itemize
+How density evolves over time (from initial distribution)
+\end_layout
+
+\begin_deeper
+\begin_layout Itemize
+Whats the peak of densities?
+ And why?
+\end_layout
+
+\begin_layout Itemize
+What's the flux of moving vehicles?
+ How does it change,
+ when is it high,
+ when low?
+ More efficient to get high distance between cars (less cars for higher speed) or short distance (more cars with less speed)?
+\end_layout
+
+\end_deeper
+\end_inset
+
 
 \end_layout
 

Project2/src/System.py

@@ -35,7 +35,6 @@ class System:
         self.NTime = NTime
         self.dt = t_end / NTime
 
-        self.A = None
         self.M = 1.
 
         # Plotting settings
@@ -144,7 +143,7 @@ class System:
             Free particles, Particles in positive direction, Particles in negative direction
         '''
         n0[0] = 0
-        n0[-1] = 3 - 2 * self.L
+        # n0[-1] = 3 - 2 * self.L
         return n0
 
     def Initialization(self) -> tuple:
@@ -169,17 +168,6 @@ class System:
 
         return x, n_init
     
-    def set_A(self, A:float) -> None:
-        '''
-        Sets the parameter A for the exponential function
-
-        Parameters
-        ----------
-        A : float
-            Parameter for the exponential function
-        '''
-        self.A = A
-    
     def f(self, u:np.array) -> np.array:
         '''
         Exponential closure for the velocity
@@ -353,7 +341,7 @@ class System:
     def plot_solution_1D(self, t:float) -> tuple:
         fig, axs = plt.subplots()
         fig.suptitle(f'Solution at t={t}')
-        axs.plot(self.x, self.n_vec[int(t * self.dt / self.NTime)], '--o')
+        axs.plot(self.x, self.n_vec[int(t / self.dt)], label='$n$')
         axs.grid()
         fig.legend()
         fig.tight_layout()