diff --git a/Project1/LyX/#Abstract.lyx# b/Project1/LyX/#Abstract.lyx#
deleted file mode 100644
index ccbdddca55fd4ca15590f76335a7f7b9835021be..0000000000000000000000000000000000000000
--- a/Project1/LyX/#Abstract.lyx#
+++ /dev/null
@@ -1,114 +0,0 @@
-#LyX 2.4 created this file. For more info see https://www.lyx.org/
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-\begin_layout Abstract
-A frequent reason for neurodegenerative diseases is the swelling of axons (nerve cells).
- This swelling is often caused by traffic jams in intracellular traffic.
- The purpose of this work is to inspect a mathematical model,
- combining diffusive and advective transport behavior,
- for intracellular traffic.
- This coupled model describes the natural movement of free particles and particles riding on microtubules towards and away from the neuron body and the interaction between these different particles.
- 
-\end_layout
-
-\begin_layout Abstract
-This work aims to under
-\end_layout
-
-\begin_layout Keywords
-Some Keywords
-\end_layout
-
-\end_body
-\end_document
diff --git a/Project1/LyX/#Implementation.lyx# b/Project1/LyX/#Implementation.lyx#
deleted file mode 100644
index 20761ee55b3282d788293738cc5e39c2e7ad342c..0000000000000000000000000000000000000000
--- a/Project1/LyX/#Implementation.lyx#
+++ /dev/null
@@ -1,268 +0,0 @@
-#LyX 2.4 created this file. For more info see https://www.lyx.org/
-\lyxformat 620
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-
-\begin_body
-
-\begin_layout Section
-Implementation
-\end_layout
-
-\begin_layout Standard
-All the following examples,
- visualizations and the discussed implementation can be found in the reproducability repository for this project (see 
-\begin_inset CommandInset citation
-LatexCommand cite
-key "Repository"
-literal "false"
-
-\end_inset
-
-).
- All the implementations are done with the Python programming language,
- mainly with the numpy package for vector operations and matplotllib for visualizations.
- Furthermore,
- there is an additional implementation using the open-source software FEniCSx
-\begin_inset Foot
-status open
-
-\begin_layout Plain Layout
-FEniCSx:
- 
-\begin_inset Flex URL
-status open
-
-\begin_layout Plain Layout
-
-https://fenicsproject.org/
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-,
- which utilizes the Finite Element Method.
- This software was used to check the Python implementation and the correctness of the method.
-\end_layout
-
-\begin_layout Subsection
-Convergence Analysis
-\end_layout
-
-\begin_layout Standard
-A convergence analysis was performed to ensure the numerical implemenation's correctness for the two uncoupled advection equations.
- The solutions for different grid sizes are compared to the analytical solution.
- To interpret the convergence of the numerical method,
- a relevant error measure is introduced.
- The 
-\begin_inset Formula $\mathcal{L}_{1}$
-\end_inset
-
- norm and its relative counterpart are calculated as
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula 
-\begin{align}
-\mathcal{L}_{1}(u) & =\int_{\Omega}\left|u-u_{\text{exact}}\right|d\Omega\\
-\mathcal{L}_{1,\text{rel}}(u) & =\frac{\mathcal{L}_{1}(u)}{\int_{\Omega}\left|u_{\text{exact}}\right|d\Omega}
-\end{align}
-
-\end_inset
-
-For the convergence analysis,
- the relative error norm 
-\begin_inset Formula $\mathcal{L}_{1,\text{rel}}$
-\end_inset
-
- will be used.
- For the convergence analysis the default parameter (see 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "subsec:ListOfVariables"
-plural "false"
-caps "false"
-noprefix "false"
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-
-\end_inset
-
-) were used.
- The grid length is chosen from 
-\begin_inset Formula $\Delta x\in\{2.0,0.2,0.02,0.002\}$
-\end_inset
-
-.
- The cfl-condition was checked for every grid size.
- If the condition was not fulfilled,
- the time step size was reduced iteratively,
- until it was fulfilled.
-\end_layout
-
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-\begin_inset Float figure
-placement document
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-
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-
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-
-\end_inset
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-
-\begin_inset Caption Standard
-
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-Convergence
-\begin_inset CommandInset label
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-
-\end_inset
-
-
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-
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-
-
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-\begin_layout Plain Layout
-
-\end_layout
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-plural "false"
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-
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-\begin_inset Formula $\mathcal{L}_{1,\text{rel}}$
-\end_inset
-
- error for both,
- the particles moving in positive and negative direction.
- Both error converge approximately with order 
-\begin_inset Formula $\frac{1}{2}$
-\end_inset
-
-.
- As the used finite difference stencil for the spatial derivative 
-\begin_inset Formula $(n_{\pm})_{x}$
-\end_inset
-
- is of first order
-\end_layout
-
-\end_body
-\end_document
diff --git a/Project1/LyX/#Introduction.lyx# b/Project1/LyX/#Introduction.lyx#
deleted file mode 100644
index de1f379ab83b447b028245135fa267afe2a2b78a..0000000000000000000000000000000000000000
--- a/Project1/LyX/#Introduction.lyx#
+++ /dev/null
@@ -1,550 +0,0 @@
-#LyX 2.4 created this file. For more info see https://www.lyx.org/
-\lyxformat 620
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-\begin_inset CommandInset label
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-name "sec:Introduction"
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Neurodegenerative diseases are a serious threat to humans health.
- One reason for these diseases is the swelling of axons,
- caused by irregularities in intracellular traffic in the the axons.
- Axons are a part of neurons,
- transmitting an electrical signal.
- These axons can be up to one meter in the human body and support little synthesis.
- Therefore,
- they need certain materials,
- imported from the cytoplasm of the cell body.
- 
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status open
-
-\begin_layout Itemize
-Neurons -> highly specialized cells with long arms (processes).
-\end_layout
-
-\begin_deeper
-\begin_layout Itemize
-axon:
- if arm transmits electrical signal
-\end_layout
-
-\begin_deeper
-\begin_layout Itemize
-up to one meter in human body
-\end_layout
-
-\begin_layout Itemize
-support little synthesis of proteins or membrane (transport equation needed)
-\end_layout
-
-\begin_deeper
-\begin_layout Itemize
-materials must be constantly imported from synthetically active cytoplasm of cell body
-\end_layout
-
-\begin_layout Itemize
-Diffusion not fast enough (Einstein:
- Brownian motion:
- large molecules diffuse slower)
-\end_layout
-
-\begin_deeper
-\begin_layout Itemize
-\begin_inset Quotes eld
-\end_inset
-
-railway system
-\begin_inset Quotes erd
-\end_inset
-
-:
- particles attach themselves to molecular motors to get transported to microtubules
-\end_layout
-
-\begin_deeper
-\begin_layout Itemize
-molecular motor:
- specialized protein that 
-\begin_inset Quotes eld
-\end_inset
-
-walks
-\begin_inset Quotes erd
-\end_inset
-
- along intracellular filaments,
- such as microtubules)
-\end_layout
-
-\end_deeper
-\end_deeper
-\end_deeper
-\end_deeper
-\begin_layout Itemize
-dendrite:
- if arm receives electrical signal
-\end_layout
-
-\begin_layout Itemize
-Polarity:
- All microtubules (MT) in axon have same polarity (plus ends point toward axon terminal)
-\end_layout
-
-\begin_deeper
-\begin_layout Itemize
-kinesin-family molecular motors:
- Transport towards plus-ends
-\end_layout
-
-\begin_layout Itemize
-dynein-family molecular motors:
- Transport to minus-ends
-\end_layout
-
-\begin_layout Itemize
-\begin_inset Quotes eld
-\end_inset
-
-Traffic jams
-\begin_inset Quotes erd
-\end_inset
-
-:
-\end_layout
-
-\begin_deeper
-\begin_layout Itemize
-occur due to irregularities in intracellular traffic in axons
-\end_layout
-
-\begin_deeper
-\begin_layout Itemize
-caused by mutations of molecular motors -> possible problems:
- swelling of axons causing various neurodegenerative diseases
-\end_layout
-
-\end_deeper
-\end_deeper
-\end_deeper
-\begin_layout Itemize
-Purpose of project:
-\end_layout
-
-\begin_deeper
-\begin_layout Itemize
-Investige transport system from paper during molecular-motor-assisted transport of intracellular organelles utilizing transport equations
-\end_layout
-
-\begin_deeper
-\begin_layout Itemize
-Learn about challenges when numerically solving advection equtaions
-\end_layout
-
-\begin_layout Itemize
-Understand transport behavior in axons
-\end_layout
-
-\begin_layout Itemize
-Identify possible dangers/problems
-\end_layout
-
-\begin_layout Itemize
-Why traffic jams occur?
- Under which circumstances?
-\end_layout
-
-\end_deeper
-\end_deeper
-\end_deeper
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Therefore,
- the problem to be solved is a coupled system of partial differential equations,
- consisting of one diffusion and two advection equations.
- It reads:
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula 
-\begin{align}
-\underbrace{\hat{n}_{\hat{t}}}_{\text{1.A}} & =\underbrace{\hat{D}_{0}\hat{n}_{\hat{x}\hat{x}}}_{\text{1.B}}-\underbrace{(1+\hat{k}_{-})\hat{n}+\hat{k}_{p}\hat{n}_{+}+\hat{k}_{n}\hat{n}_{-}}_{\text{1.C}}\label{eq:Dimensions_n}\\
-\underbrace{\left(\hat{n}_{+}\right)_{\hat{t}}}_{\text{2.A}}+\underbrace{\left(\frac{V_{+}}{V_{+,0}}\hat{n}_{+}\right)_{\hat{x}}}_{\text{2.B}} & =\underbrace{\hat{k_{+}}\hat{n}-\hat{k}_{p}\hat{n}_{+}}_{\text{2.C}}\qquad\hat{x}\in[0,\hat{L}]\label{eq:Dimensions_n+}\\
-\underbrace{\left(\hat{n}_{-}\right)_{\hat{t}}}_{\text{3.A}}+\underbrace{\left(\frac{V_{-}}{V_{+,0}}\hat{n}_{-}\right)_{\hat{x}}}_{\text{3.B}} & =\underbrace{\hat{k}_{-}\hat{n}-\hat{k}_{n}\hat{n}_{-}}_{\text{3.C}}\label{eq:Dimensions_n-}
-\end{align}
-
-\end_inset
-
-with free particle concentration 
-\begin_inset Formula $\hat{n}$
-\end_inset
-
- and 
-\begin_inset Formula $\hat{n}_{\pm}$
-\end_inset
-
- the concentration of particles moving in positive/negative direction.
- The spatial domain is described by 
-\begin_inset Formula $\Omega=[0,L]$
-\end_inset
-
- on a timeframe of 
-\begin_inset Formula $[0,t_{\text{end}}]$
-\end_inset
-
-.
- Furthermore,
- 
-\begin_inset Formula $\hat{D}_{0}$
-\end_inset
-
- is the diffusion coefficient,
- 
-\begin_inset Formula $V_{\pm}$
-\end_inset
-
- the molecular motor velocity along microtubules regarding the 
-\begin_inset Formula $\pm$
-\end_inset
-
- direction,
- 
-\begin_inset Formula $V_{\pm,0}$
-\end_inset
-
- molecular velocity along microtubules regarding 
-\begin_inset Formula $\pm$
-\end_inset
-
- direction in the case that the concentration of molecules riding on microtubules is very low,
- the rate coefficients 
-\begin_inset Formula $\hat{k}_{p},\hat{k}_{n},\hat{k}_{+}$
-\end_inset
-
- and 
-\begin_inset Formula $\hat{k}_{-}$
-\end_inset
-
-.
- Equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:Dimensions_n"
-plural "false"
-caps "false"
-noprefix "false"
-nolink "false"
-
-\end_inset
-
- describes the movement from the free particles by diffusion.
- The temporal change is described by term 1.A and the diffusive behavior by 1.B.
- The source term 1.C describes the exchange of particles with 
-\begin_inset Formula $\hat{n}_{\pm}$
-\end_inset
-
-.
- The two advection equations 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:Dimensions_n+"
-plural "false"
-caps "false"
-noprefix "false"
-nolink "false"
-
-\end_inset
-
- and 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:Dimensions_n-"
-plural "false"
-caps "false"
-noprefix "false"
-nolink "false"
-
-\end_inset
-
- describe the movement of the particles moving in positive and negative direction due to convection.
- The terms 2.A and 3.A describe the temporal change,
- and 2.B and 3.B the advective behavior.
- Both equations also have some exchange of concentration with the other particles,
- described by terms 2.C and 3.C.
-\end_layout
-
-\begin_layout Standard
-Equations 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:Dimensions_n+"
-plural "false"
-caps "false"
-noprefix "false"
-nolink "false"
-
-\end_inset
-
- and 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:Dimensions_n-"
-plural "false"
-caps "false"
-noprefix "false"
-nolink "false"
-
-\end_inset
-
- can be reformulated into a generic advection equation of the form
-\begin_inset Formula 
-\begin{equation}
-u_{t}+a(x,t)u_{x}=q(x,t,u)\label{eq:GenericAdvectionEquation}
-\end{equation}
-
-\end_inset
-
- with the advection velocity 
-\begin_inset Formula $a$
-\end_inset
-
-,
- the property of interest 
-\begin_inset Formula $u$
-\end_inset
-
- and the source term 
-\begin_inset Formula $q$
-\end_inset
-
-.
- This results in
-\begin_inset Formula 
-\begin{align}
-u_{t,2} & =\left(\hat{n}_{+}\right)_{\hat{t}},\quad a_{2}(x,t)=\frac{V_{+}}{V_{+,0}},\quad u_{x,2}=\hat{n}_{+,\hat{x}},\quad q_{3}(x,t,u)=\hat{k}_{p}\hat{n}-\hat{k}_{p}\hat{n}_{+}\\
-u_{t,3} & =\left(\hat{n}_{-}\right)_{\hat{t}},\quad a_{3}(x,t)=\frac{V_{-}}{V_{-,0}},\quad u_{x,3}=\hat{n}_{-,\hat{x}},\quad q_{3}(x,t,u)=\hat{k}_{-}\hat{n}-\hat{k}_{n}\hat{n}_{-}
-\end{align}
-
-\end_inset
-
-Therefore,
- it is sufficient to know how to deal with a generic advection equation in the form of equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:GenericAdvectionEquation"
-plural "false"
-caps "false"
-noprefix "false"
-nolink "false"
-
-\end_inset
-
- to solve the two advection equations,
- describing the particle transport in axons,
- both analytically and numerically.
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-Engineering problem:
-\end_layout
-
-\begin_layout Itemize
-lets do some stuff
-\end_layout
-
-\begin_layout Plain Layout
-Mathematical model:
-\end_layout
-
-\begin_layout Itemize
-lets do some stuff
-\end_layout
-
-\begin_layout Plain Layout
-Purpose of each transport equation,
- role of each term:
-\end_layout
-
-\begin_layout Itemize
-lets do some stuff
-\end_layout
-
-\begin_layout Plain Layout
-Few relevant research questions
-\end_layout
-
-\begin_layout Itemize
-lets do some stuff
-\end_layout
-
-\begin_layout Plain Layout
-Why need to know how to deal with generic advection equation?
- (
-\begin_inset Formula $u_{t}+a(x,t)u_{x}=q(x,t,u)$
-\end_inset
-
-
-\end_layout
-
-\begin_layout Itemize
-lets do some stuff
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Float figure
-placement document
-alignment document
-wide false
-sideways false
-status open
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-	filename Figures/Schematic.jpg
-	width 30text%
-
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-
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-
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diff --git a/Project1/LyX/#Results.lyx# b/Project1/LyX/#Results.lyx#
deleted file mode 100644
index 207080feeb363f36469905afb83adfa91cd09452..0000000000000000000000000000000000000000
--- a/Project1/LyX/#Results.lyx#
+++ /dev/null
@@ -1,1024 +0,0 @@
-#LyX 2.4 created this file. For more info see https://www.lyx.org/
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-\end_inset
-
-Inserting the dimensionless quantities in equations 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:Dimensions_n"
-plural "false"
-caps "false"
-noprefix "false"
-nolink "false"
-
-\end_inset
-
-,
- 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:Dimensions_n+"
-plural "false"
-caps "false"
-noprefix "false"
-nolink "false"
-
-\end_inset
-
- and 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:Dimensions_n-"
-plural "false"
-caps "false"
-noprefix "false"
-nolink "false"
-
-\end_inset
-
- under the additional assumption of constant velocities (
-\begin_inset Formula $V_{+}=V_{+,0},V_{-}=V_{-,0}=-V_{+,0})$
-\end_inset
-
-,
- yields the dimensionless system of equations
-\begin_inset Formula 
-\begin{align}
-n_{t} & =D_{0}n_{xx}-\left(1+k_{-}\right)n+k_{p}n_{+}+k_{n}n_{-}\label{eq:Dimensionless_n}\\
-\left(n_{+}\right)_{t}+\left(n_{+}\right)_{x} & =k_{+}n-k_{p}n_{+}\qquad x\in[0,L]\label{eq:Dimensionless_n_+}\\
-\left(n_{-}\right)_{t}-\left(n_{-}\right)_{x} & =k_{-}n-k_{n}n_{-}\label{eq:Dimensionless_n_-}
-\end{align}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-Initial- and Boundary Conditions
-\end_layout
-
-\begin_layout Standard
-For the initial state a linear relation for the free particles and a vanishing amount of particles moving along one direction is used
-\begin_inset Formula 
-\begin{align}
-n(x,t=0) & =n^{0}(x)=+\left(N_{L}-N_{0}\right)\frac{x}{L}\\
-n_{+}(x,t=0) & =n_{+}^{0}(x)=0\\
-n_{-}(x,t=0) & =n_{-}^{0}(x)=0
-\end{align}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-Different Hypothesis's for Advective Transport
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-To Do
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-Method of Characteristics
-\begin_inset CommandInset label
-LatexCommand label
-name "subsec:Method-of-Characteristics"
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-The method of characteristics (MOC) can be utilized to solve an advection equation in terms of an initial-value problem.
- Uncoupling equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:Dimensionless_n_+"
-plural "false"
-caps "false"
-noprefix "false"
-nolink "false"
-
-\end_inset
-
- from the remaining equations results in
-\begin_inset Formula 
-\begin{align}
-\left(n_{+}\right)_{t}+\left(n_{+}\right)_{x} & =0\\
-\Leftrightarrow U_{t}+U_{x} & =0\label{eq:Characteristic_UtUx}\\
-\text{with: }U(x,0) & =U_{0}(x)\quad\forall x\in\mathbb{R}
-\end{align}
-
-\end_inset
-
-As an ansatz assume that some curve 
-\begin_inset Formula $x(t)$
-\end_inset
-
- along the slope of 
-\begin_inset Formula $U$
-\end_inset
-
- is constant,
- meaning
-\begin_inset Formula 
-\begin{align}
-0 & =\frac{d}{dt}U\left(x(t),t\right)\\
- & =U_{t}\left((x(t),t\right)+U_{x}\left(x(t),t\right)\frac{dx}{dt}
-\end{align}
-
-\end_inset
-
-As 
-\begin_inset Formula $U_{t}+U_{x}=0$
-\end_inset
-
- (see equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:Characteristic_UtUx"
-plural "false"
-caps "false"
-noprefix "false"
-nolink "false"
-
-\end_inset
-
-) 
-\begin_inset Formula $\frac{dx}{dt}$
-\end_inset
-
- has to be equal to one to fulfill the partial differential equation.
- Insert this with
-\begin_inset Formula 
-\begin{align}
-\frac{dx}{dt} & =1\quad\text{with: }x(0)=x_{0}\\
-\Rightarrow x(t) & =x_{0}+t
-\end{align}
-
-\end_inset
-
-resulting in the solution 
-\begin_inset Formula $U(x,t)=U_{0}(x_{0})=U_{0}(x-t)$
-\end_inset
-
- and for the concentrations
-\begin_inset Formula 
-\begin{equation}
-n_{+}(x,t)=n_{+,0}(x-t)
-\end{equation}
-
-\end_inset
-
-The analytical solution for the uncoupled equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:Dimensionless_n_-"
-plural "false"
-caps "false"
-noprefix "false"
-nolink "false"
-
-\end_inset
-
- can be derived in the same way,
- resulting in 
-\begin_inset Formula $n_{-}(x,t)=n_{-,0}(x+t)$
-\end_inset
-
-.
- Note that the advection velocity is now 
-\begin_inset Formula $a=-1$
-\end_inset
-
-,
- therefore 
-\begin_inset Formula $\frac{dx}{dt}=-1$
-\end_inset
-
- has to be fulfilled.
- 
-\end_layout
-
-\begin_layout Subsection
-Discretization with Finite Differences and Upwind Discretization
-\end_layout
-
-\begin_layout Standard
-The coupled system of equations will be solved numerically with the finite-difference scheme and an upwind-discretization for the advective term.
- In the discrete terms,
- the upper index 
-\begin_inset Formula $\cdot^{j}$
-\end_inset
-
- indicates the 
-\begin_inset Formula $j$
-\end_inset
-
--th time step,
- while the lower index 
-\begin_inset Formula $\cdot_{i}$
-\end_inset
-
- indicates the i-th grid point.
- 
-\begin_inset Formula $\Delta t$
-\end_inset
-
- is the time-step size and 
-\begin_inset Formula $\Delta x$
-\end_inset
-
- the distant between two grid points on an uniform grid.
-\end_layout
-
-\begin_layout Standard
-Discretizing the time with a simple explicit Euler method,
- the diffusion coefficient with a finite difference stencil of second order and the advective terms with an Upwind discretization,
- the discretized equations are as follows
-\begin_inset Formula 
-\begin{align}
-\frac{n_{i}^{j+1}-n_{i}^{j}}{\Delta t} & =D_{0}\frac{n_{i+1}^{j}-2n_{i}^{j}+n_{i-1}^{j}}{(\Delta x)^{2}}-(1+k_{-})n_{i}^{j}+k_{p}n_{+,i}^{j}+k_{n}n_{-,i}^{j}\label{eq:Discretized_1}\\
-\frac{n_{+,i}^{j+1}-n_{+,i}^{j}}{\Delta t}+\frac{1}{\Delta x}\left(U_{+,i+\frac{1}{2}}^{j}-U_{+,i-\frac{1}{2}}^{j}\right) & =k_{-}n_{i}^{j}-k_{p}n_{+,i}^{j}\label{eq:Discretized_2}\\
-\frac{n_{-,i}^{j+1}-n_{-,i}^{j}}{\Delta t}-\frac{1}{\Delta x}\left(U_{-,i+\frac{1}{2}}^{j}-U_{-,i-\frac{1}{2}}^{j}\right) & =k_{-}n_{i}^{j}-k_{n}n_{-,i}^{j}\label{eq:Discretized_3}
-\end{align}
-
-\end_inset
-
-with 
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-Check this
-\end_layout
-
-\end_inset
-
-
-\begin_inset Formula 
-\begin{align}
-U_{+,i+\frac{1}{2}}^{j} & =n_{+,i}^{j},\quad U_{+,i-\frac{1}{2}}^{j}=n_{+,i-1}^{j}\label{eq:Upwind_+}\\
-U_{-,i+\frac{1}{2}}^{j} & =n_{-,i+1}^{j},\quad U_{-,i-\frac{1}{2}}^{j}=n_{-,i}^{j}\label{eq:Upwind_-}
-\end{align}
-
-\end_inset
-
-Inserting the upwind formulations into the discretized system of equations (
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:Discretized_1"
-plural "false"
-caps "false"
-noprefix "false"
-nolink "false"
-
-\end_inset
-
--
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:Discretized_3"
-plural "false"
-caps "false"
-noprefix "false"
-nolink "false"
-
-\end_inset
-
-) and reformulating this yields an explicit scheme for time-step 
-\begin_inset Formula $j+1$
-\end_inset
-
-,
- depending on time-step 
-\begin_inset Formula $j$
-\end_inset
-
-.
-\begin_inset Formula 
-\begin{align}
-n_{i}^{j+1} & =n_{i}^{j}+\Delta t\left(D_{0}\frac{n_{i+1}^{j}-2n_{i}^{j}+n_{i-1}^{j}}{(\Delta x)^{2}}-(1+k_{-})n_{i}^{j}+k_{p}n_{+,i}^{j}+k_{n}n_{-,i}^{j}\right)\label{eq:Discretized_Explicit_1}\\
-n_{+,i}^{j+1} & =n_{-,i}^{j}+\Delta t\left(-\frac{1}{\Delta x}\left(n_{+,i}^{j}-n_{+,i-1}^{j}\right)+k_{-}n_{i}^{j}-k_{p}n_{+,i}^{j}\right)\\
-n_{-,i}^{j+1} & =n_{-,i}^{j}+\Delta t\left(+\frac{1}{\Delta x}\left(n_{-,i+1}^{j}-n_{-,i}^{j}\right)+k_{-}n_{i}^{j}-k_{n}n_{-,i}^{j}\right)\label{eq:Discretized_Explicit_3}
-\end{align}
-
-\end_inset
-
-which is straight forward to implement in Python with the numpy library,
- in the case of homogeneous Neumann boundary conditions,
- as they will be used in this work.
- A for-loop over all time steps has to be implemented and for each time-step the explicit update,
- following equations 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:Discretized_Explicit_1"
-plural "false"
-caps "false"
-noprefix "false"
-nolink "false"
-
-\end_inset
-
--
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:Discretized_Explicit_3"
-plural "false"
-caps "false"
-noprefix "false"
-nolink "false"
-
-\end_inset
-
-,
- has to be called.
- After this,
- the Dirichlet boundary conditions can be enforced by setting the boundary points 
-\begin_inset Formula $n_{0}^{j+1},n_{m}^{j+1},n_{+,0}^{j+1},n_{-,m}^{j+1}$
-\end_inset
-
- to their respective value.
-\end_layout
-
-\begin_layout Subsection
-Stability
-\end_layout
-
-\begin_layout Standard
-As the numerical scheme is of an explicit form,
- it is not unconditionally stable.
- For the 
-\begin_inset Formula $L^{1}$
-\end_inset
-
--estimate,
- to fulfill conservation of the concentrations 
-\begin_inset Formula $n,n_{+},n_{-}$
-\end_inset
-
-,
- the scheme has to fulfill
-\begin_inset Formula 
-\begin{equation}
-\int_{0}^{1}|u(x,t)|dx\leq\int_{0}^{1}|u_{0}(x)|dx
-\end{equation}
-
-\end_inset
-
-or in the discretized formulation
-\begin_inset Formula 
-\begin{equation}
-\sum_{j=1}^{M}|u_{j}^{n+1}|\Delta x\leq\sum_{j=1}^{M}|u_{0,j}|\Delta x\quad\Leftrightarrow\sum_{j=1}^{M}|u_{j}^{n+1}|\leq\sum_{j=1}^{n}|u_{j}^{n}|\label{eq:Stability_Constraint}
-\end{equation}
-
-\end_inset
-
-For the two uncoupled advection equations under the assumptions of a uniform advection velocity 
-\begin_inset Formula $a$
-\end_inset
-
-=1 this results in the following derivation.
- 
-\end_layout
-
-\begin_layout Standard
-With the triangular inequality (
-\begin_inset Formula $|a+b|\leq|a|+|b|$
-\end_inset
-
-) and 
-\begin_inset Formula $\lambda=\frac{\Delta t}{\Delta x}$
-\end_inset
-
- the concentration at the next time step can be reformulated to
-\begin_inset Formula 
-\begin{equation}
-\left|u_{j}^{n+1}\right|u_{j}^{n}-\lambda(u_{j}^{n}-u_{j-1}^{n})\leq(1-\lambda)\left|u_{j}^{n}\right|+\lambda\left|u_{j-1}^{n}\right|
-\end{equation}
-
-\end_inset
-
-under the constraint 
-\begin_inset Formula $1-\lambda\geq0$
-\end_inset
-
-.
- Tis discretized concentration can be inserted into equation 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:Stability_Constraint"
-plural "false"
-caps "false"
-noprefix "false"
-nolink "false"
-
-\end_inset
-
- to get
-\begin_inset Formula 
-\begin{align}
-\sum_{j=1}^{M}\left|u_{j}^{n+1}\right| & \leq(1-\lambda)\sum_{j=1}^{M}\left|u_{j}^{n}\right|+\lambda\sum_{j=1}^{M}\left|u_{j-1}^{n}\right|\\
- & =(1-\lambda)\sum_{j=1}^{M}\left|u_{j}^{n}\right|+\lambda\sum_{j=1}^{M}\left|u_{j}^{n}\right|\\
- & =\sum_{j=1}^{M}\left|u_{j}^{n}\right|
-\end{align}
-
-\end_inset
-
-Therefore,
- the discretization for the advection equation is stable under the condition
-\begin_inset Formula 
-\begin{align}
-1-\lambda & =1-\frac{\Delta t}{\Delta x}\geq0\\
-\Leftrightarrow\frac{\Delta t}{\Delta x} & \leq1
-\end{align}
-
-\end_inset
-
-For general advection equation (
-\begin_inset Formula $a=a)$
-\end_inset
-
- the stability constraint is called the CFL-condition and reads:
-\begin_inset Formula 
-\begin{equation}
-a\frac{\Delta t}{\Delta x}\leq1
-\end{equation}
-
-\end_inset
-
-For the diffusion equation a similar derivation stability can be performed,
- resulting in the more strict constraint
-\begin_inset Formula 
-\begin{equation}
-D_{0}\frac{\Delta t}{\left(\Delta x\right)^{2}}\leq\frac{1}{2}
-\end{equation}
-
-\end_inset
-
-
-\end_layout
-
-\end_body
-\end_document