diff --git a/Project2/LyX/DiscussionAndConclusion.lyx b/Project2/LyX/DiscussionAndConclusion.lyx
index 94a813ae36af5deab172ea96e53d06b5cd7c308e..d8aa73fa3e0e6ebc0a78fbad9991fd03d3ad1ffb 100644
--- a/Project2/LyX/DiscussionAndConclusion.lyx
+++ b/Project2/LyX/DiscussionAndConclusion.lyx
@@ -155,12 +155,14 @@ If a traffic jam occurs,
there is a simple approach to dissolve it,
again.
The cars should hold a longer distance to the car ahead,
- to make sure that the density of cars at the end of the traffic flow is smaller then before.
+ to make sure that the density of cars at the end of the traffic flow is smaller then before,
+ increasing the flux function.
With this,
a rarefaction wave will build and the traffic flow will be dissolved over time.
If cars hold a short distance to the cars in front,
- the density at the end of the traffic jam will probably be higher then before,
- resulting in a certain increase in the significance of the traffic jam.
+ the density at the end of the traffic jam will probably be higher then in the beginning,
+ leading to a smaller flux function at the end of the traffic flow,
+ resulting in more cars entering the traffic flow than leaving it.
\end_layout
\begin_layout Standard
@@ -189,11 +191,7 @@ Use a non-linear model for the velocity of cars.
\begin_layout Enumerate
It is unrealistic,
that the velocity of cars is linearly proportional to the density of cars.
-
-\end_layout
-
-\begin_layout Enumerate
-In a more realistic scenario,
+ In a more realistic scenario,
the velocity of cars would decrease faster for higher densities and slower for lower densities.
\end_layout
diff --git a/Project2/LyX/Figures/Convergence_abs.pdf b/Project2/LyX/Figures/Convergence_abs.pdf
index 5ed3fdfe12e6115ff81d6096e3e4e4eaf4971316..c1f76eed9552708aed931cc5817f14298c5cb779 100644
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diff --git a/Project2/LyX/Figures/Convergence_rel.pdf b/Project2/LyX/Figures/Convergence_rel.pdf
index 72c34b15a4a4c8c6b64356a50610bbcb8218af97..6291fd4bffd334c9f438bf5c92ffb76c6a5f4dfd 100644
Binary files a/Project2/LyX/Figures/Convergence_rel.pdf and b/Project2/LyX/Figures/Convergence_rel.pdf differ
diff --git a/Project2/LyX/Figures/Problem_2_3_1D.pdf b/Project2/LyX/Figures/Problem_2_3_1D.pdf
index c49bd15f24d04db24cac6f5f31d254f2dba622cf..9ede8c6e4e30f802c8017dc027c4b89896c8b206 100644
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diff --git a/Project2/LyX/Figures/Problem_2_3_3D.pdf b/Project2/LyX/Figures/Problem_2_3_3D.pdf
index e816679a96453a95143f973766b109b83754e36a..f90743f7697ab83c6d7a7ef835e9262d43ade1cc 100644
Binary files a/Project2/LyX/Figures/Problem_2_3_3D.pdf and b/Project2/LyX/Figures/Problem_2_3_3D.pdf differ
diff --git a/Project2/LyX/Figures/Problem_2_3_abs_error.pdf b/Project2/LyX/Figures/Problem_2_3_abs_error.pdf
index 62b80b23d17222c947e061bb063c71ebac807d3c..5eb0680d292c73ee21a1fb1212ec00c7058a033e 100644
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diff --git a/Project2/LyX/Implementation.lyx b/Project2/LyX/Implementation.lyx
index 02dcbeb7f56196b82b44c3051fb32a1ed02f2e16..5b2d6a1ce7fad54da8b016222b1969cfe581f1ba 100644
--- a/Project2/LyX/Implementation.lyx
+++ b/Project2/LyX/Implementation.lyx
@@ -90,6 +90,13 @@
\begin_layout Section
Implementation
+\begin_inset CommandInset label
+LatexCommand label
+name "sec:Implementation"
+
+\end_inset
+
+
\end_layout
\begin_layout Subsection
@@ -99,7 +106,29 @@ Discretization with Finite Differences and Rusanov Discretization
\begin_layout Standard
The partial differential equation will be solved numerically with the finite-difference scheme and a Rusanov discretization,
which adds a small amount of artificial diffusion to stabilize the advective term.
- In the discrete terms,
+
+\begin_inset Formula
+\begin{align}
+u_{t}+f(u)_{x} & =0\\
+\Rightarrow u_{t}^{\epsilon}+f(u_{t}^{\epsilon})_{x} & =u_{xx}^{\epsilon}
+\end{align}
+
+\end_inset
+
+where
+\begin_inset Formula $u^{\epsilon}$
+\end_inset
+
+ is the solution with the added diffusion.
+ The upper index
+\begin_inset Formula $\epsilon$
+\end_inset
+
+ will be dropped for more readability.
+\end_layout
+
+\begin_layout Standard
+In the discrete terms,
the upper index
\begin_inset Formula $\cdot^{j}$
\end_inset
@@ -123,10 +152,7 @@ The partial differential equation will be solved numerically with the finite-dif
\end_inset
the distance between two grid points on an uniform grid.
-\end_layout
-
-\begin_layout Standard
-Discretizing the time with a simple explicit Euler method and the space with the aforementioned Rusanov discretization yields the discretized formulation.
+ Discretizing the time with a simple explicit Euler method and the space with the aforementioned Rusanov discretization yields the discretized formulation.
\begin_inset Formula
\begin{align}
\frac{u_{i}^{j+1}-u_{i}^{j}}{\Delta t} & +\frac{1}{\Delta x}\left(F_{i+\frac{1}{2}}^{j}-F_{i-\frac{1}{2}}^{j}\right)=0\\
@@ -136,20 +162,15 @@ M & \sim\max\left|f'\right|
\end_inset
-Reformulating this to
+Reformulate this to
\begin_inset Formula $u_{i}^{j+1}$
\end_inset
- yields an explicit scheme for time-step
+ to get an explicit scheme for time-step
\begin_inset Formula $j+1$
\end_inset
-,
- depending only on time-step
-\begin_inset Formula $j$
-\end_inset
-
-.
+ results in
\begin_inset Formula
\begin{equation}
u_{i}^{j+1}=u_{i}^{j}-\frac{\Delta t}{\Delta x}\left(F_{i+\frac{1}{2}}^{j}-F_{i-\frac{1}{2}}^{i}\right)
@@ -166,7 +187,7 @@ The CFL-Condition as a Stability Criteria
\begin_layout Standard
To ensure numerical stability,
- the cfl-condion has to hold true,
+ the cfl-condition has to hold true,
which reads
\begin_inset Formula
\begin{equation}
@@ -183,7 +204,8 @@ To ensure numerical stability,
\begin_inset Formula $f(u)=a(u)u$
\end_inset
-.
+ (for more information,
+ see Project 1).
\end_layout
\begin_layout Subsection
@@ -227,7 +249,7 @@ A convergence analysis was performed to ensure the numerical implementation's co
norm and its relative counterpart
\begin_inset Formula
\begin{align}
-\mathcal{L}_{1}(u) & =\int\left|u-u_{\text{exact}}\right|d\Omega\\
+\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}
@@ -259,7 +281,7 @@ u(x,t)=\begin{cases}
\end_inset
-For the derivation of the analytical solution see Section
+For the derivation of the analytical solution see the upcoming Section
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:AnalyticalSolution_2.3"
@@ -271,18 +293,6 @@ nolink "false"
\end_inset
.
-
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-ToDo:
- Add a reference to where the analytical solution is computed
-\end_layout
-
-\end_inset
-
-
\end_layout
\begin_layout Standard
@@ -311,7 +321,7 @@ status open
\begin_layout Plain Layout
The numerical scheme converges with an order somewhat below first order.
- The x-axis shows the number of cells and the y-axis the
+ The x-axis shows the cell size and the y-axis the
\begin_inset Formula $\mathcal{L}_{1,\text{rel}}$
\end_inset
@@ -342,7 +352,7 @@ The convergence was inspected on a domain
\end_inset
with
-\begin_inset Formula $\Delta x\in[10,100,1.000,10.000]$
+\begin_inset Formula $\Delta x\in[0.7,0.07,0.007,0.0007]$
\end_inset
).
@@ -366,7 +376,7 @@ nolink "false"
\end_inset
- indicates the the numerical scheme converges with an order somewhat below first order.
+ indicates that the numerical scheme converges with an order somewhat below first order.
Therefore,
a refined grid results in a more accurate solution.
For
diff --git a/Project2/LyX/Introduction.lyx b/Project2/LyX/Introduction.lyx
index 0713711b8884ea22e0e382c3e186650e6ddfe31c..bab0d9a3b5d32766afa83324403808bcd21cf594 100644
--- a/Project2/LyX/Introduction.lyx
+++ b/Project2/LyX/Introduction.lyx
@@ -104,9 +104,9 @@ 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.
+ to get to their job,
+ meet friends,
+ do grocery shopping 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,
@@ -114,19 +114,9 @@ Traffic jams occur in our everyday life.
\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,
+The arising question is,
+ how to reduce these traffic jams to increase the quality of these.
+ This work investigates the underlying mathematical model of traffic jams,
which states a smaller vehicle speed at higher number densities.
\begin_inset Float figure
placement document
@@ -147,7 +137,9 @@ status open
\begin_inset Caption Standard
\begin_layout Plain Layout
-Image by Al Gг from Pixabay
+High number densities of cars lead to a smaller speed of those,
+ resulting in traffic jams.
+ Image by Al Gг from Pixabay
\end_layout
\end_inset
@@ -159,7 +151,7 @@ Image by Al Gг from Pixabay
\begin_inset Note Note
-status open
+status collapsed
\begin_layout Plain Layout
Engineering Problem:
@@ -274,7 +266,7 @@ where
\begin_inset Formula $V(u)$
\end_inset
- is the velocity of vehicles,
+ the velocity of vehicles,
depending on the number density and
\begin_inset Formula $b-a$
\end_inset
@@ -291,6 +283,13 @@ u\big|_{x=a}=u_{\text{in}}
is assumed to hold true.
This inflow boundary condition models the number of arriving cars at the beginning of the road.
+ In this work,
+
+\begin_inset Formula $u_{\text{in}}=u(x=x_{\text{in}})$
+\end_inset
+
+ will be utilized,
+ making it consistent with the initial distribution.
\end_layout
\begin_layout Standard
@@ -306,16 +305,11 @@ u\big|_{x=x_{\text{in}}} & =u_{\text{in}}
\end_inset
-In this work,
-
-\begin_inset Formula $u_{\text{in}}$
-\end_inset
-
- will be used as
-\begin_inset Formula $u_{0}(x=x_{\text{in}})$
+with the flux function
+\begin_inset Formula $f(u)$
\end_inset
- making it consistent with the initial distribution.
+.
The main assumption for this model is,
that the vehicle velocity depends on the number density
\begin_inset Formula
@@ -328,24 +322,16 @@ V(u)\propto(1-u)
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]\\
+\Rightarrow u_{t}+\left((1-u)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}
@@ -361,7 +347,7 @@ Arising Research Questions
\end_layout
\begin_layout Standard
-From this mathematical model and the physical background some research questions arise.
+From the mathematical model and the physical background some research questions arise.
\end_layout
\begin_layout Enumerate
@@ -387,9 +373,67 @@ Is it more efficient to have a high distance or short distance between the cars?
but less cars are on the road.
\end_layout
+\begin_layout Subsection
+Format of this Work
+\end_layout
+
+\begin_layout Standard
+Section
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sec:Theory-and-Methods"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+ will introduce the theoretical background to solve the aforementioned general conservation law analytically.
+ In Section
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sec:Implementation"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+ a method to solve the equation numerically,
+ will be introduced and Section
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sec:Results"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+ will show different examples for both,
+ the analytical and numerical,
+ solution.
+ Finally,
+ Section
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sec:Discussion-and-Conclusion"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+ will discuss the results and gives a small outlook.
+\end_layout
+
\begin_layout Standard
\begin_inset Note Note
-status open
+status collapsed
\begin_layout Plain Layout
Relevant research questions
diff --git a/Project2/LyX/Results.lyx b/Project2/LyX/Results.lyx
index 8078ebb75770dcc77d0dac54e837a368d068d90f..c1dd0f49c2c39f01eac0da3ebe70fd8ac23f4856 100644
--- a/Project2/LyX/Results.lyx
+++ b/Project2/LyX/Results.lyx
@@ -106,12 +106,75 @@ name "sec:Results"
\end_layout
+\begin_layout Standard
+In the following Section different examples will be investigated with both,
+ analytical and numerical,
+ solutions.
+ Section
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "subsec:Formation-of-Discontinuities"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+ shows the analytical calculation of the time of a discontinuity,
+ Section
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "subsec:A-Second-Model-Problem"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+ shows the analytical and numerical solution for a more complex problem,
+ which is still a model problem.
+ Finally,
+ Sections
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "subsec:ConnectionToTrafficFlowProblem"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+ (analytical and numerical) and
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "subsec:A-Second-Traffic-Flow-Problem"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+ (numerical) solve a problem,
+ related to traffic flows.
+\end_layout
+
\begin_layout Subsection
Formation of Discontinuities for the convex flux function
\begin_inset Formula $f(u)=u^{2}$
\end_inset
+\begin_inset CommandInset label
+LatexCommand label
+name "subsec:Formation-of-Discontinuities"
+
+\end_inset
+
+
\end_layout
\begin_layout Standard
@@ -142,14 +205,10 @@ u_{0}(x)=\begin{cases}
\end_inset
-
-\end_layout
-
-\begin_layout Standard
-For this example,
- it is interesting to calculate the time when a discontinuity is first formed.
- For this,
- equation
+It is interesting to calculate the time,
+ when a discontinuity occurs,
+ to analyze the breaking time of the analytical solution.
+ Equation
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:FormationDiscontinuities"
@@ -160,7 +219,7 @@ nolink "false"
\end_inset
- will be used.
+ will be used for this.
\end_layout
\begin_layout Standard
@@ -187,11 +246,15 @@ On the right part of the domain (
\end_inset
-Conclude from this that for time
+Conclude from this that the solution is valid for
+\begin_inset Formula $t<\frac{1}{4}$
+\end_inset
+
+ and for
\begin_inset Formula $t=\frac{1}{4}$
\end_inset
- a first shock forms.
+ a first discontinuity forms.
\end_layout
\begin_layout Subsection
@@ -200,6 +263,13 @@ A Second Model Problem with
\end_inset
+\begin_inset CommandInset label
+LatexCommand label
+name "subsec:A-Second-Model-Problem"
+
+\end_inset
+
+
\end_layout
\begin_layout Standard
@@ -224,15 +294,7 @@ u_{0}(x)=\begin{cases}
2x & ,x\in(0,\frac{1}{2}]\\
1 & ,x\in(\frac{1}{2},1]\\
3-2x & ,x\in(1,\frac{3}{2}]
-\end{cases}
-\end{equation}
-
-\end_inset
-
-and the derivative of the initial state
-\begin_inset Formula
-\begin{equation}
-u_{0}'(x)=\begin{cases}
+\end{cases}\qquad u_{0}'(x)=\begin{cases}
2 & ,x\in(0,\frac{1}{2}]\\
0 & ,x\in(\frac{1}{2},1]\\
-2 & ,x\in(1,\frac{3}{2}]
@@ -245,7 +307,7 @@ u_{0}'(x)=\begin{cases}
\end_layout
\begin_layout Standard
-For the solution the steps described in section
+The steps from section
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Theory-and-Methods"
@@ -256,7 +318,7 @@ nolink "false"
\end_inset
- will be utilized.
+ will be utilized to find an analytical solution.
First,
calculate the characteristic with
\begin_inset Formula
@@ -270,7 +332,7 @@ x & =x_{0}+tu_{0}(x_{0})\\
& =\begin{cases}
x_{0}\left(1+2t\right) & ,0<x_{0}\leq\frac{1}{2}\\
x_{0}+t & ,\frac{1}{2}<x_{0}\leq1\\
-x_{0}\left(1-2t\right)+3t & ,\frac{1}{2}<x_{0}\leq1
+x_{0}\left(1-2t\right)+3t & ,1<x_{0}\leq\frac{3}{2}
\end{cases}\\
\Leftrightarrow x_{0} & =\begin{cases}
\frac{x}{1+2t} & ,0<x_{0}=\frac{x}{1+2t}\leq\frac{1}{2}\\
@@ -281,8 +343,7 @@ x-t & ,\frac{1}{2}<x_{0}=x-t\leq1\\
\end_inset
-In the second step,
- the solution will be calculated based on formula
+The final solution can be found with formula
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:GeneralAnalyticalSolution"
@@ -413,6 +474,13 @@ Visualization of the solution of the scalar PDE and its flux function over space
initially there are some point-wise discrepancies,
shown by the absolute error,
indicating problems of the numerical implementation.
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:A-Second-Model-Problem"
+
+\end_inset
+
+
\end_layout
\end_inset
@@ -429,6 +497,49 @@ Visualization of the solution of the scalar PDE and its flux function over space
\end_layout
+\begin_layout Standard
+Figure
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:A-Second-Model-Problem"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+ visualizes the solution to the problem over space and time.
+ It can be seen,
+ that the numerical solution is an almost perfect match with the analytical solution.
+ However,
+ initially there are some point-wise discrepancies between both solutions,
+ indicating some problems with the numerical solution.
+ Reasons for this can be the choice of the artificial diffusion,
+ defined by
+\begin_inset Formula $M$
+\end_inset
+
+.
+
+\end_layout
+
+\begin_layout Standard
+The initial distribution slowly dissolves towards the inflow value
+\begin_inset Formula $u_{\text{in}}=0$
+\end_inset
+
+,
+ faster at the side of the inflow condition,
+ than the other side.
+ The flux function behaves similar but more smooth,
+ as the quadratic flux function increases non-linear with the property
+\begin_inset Formula $u$
+\end_inset
+
+.
+\end_layout
+
\begin_layout Subsection
Creating a Connection to the Traffic Flow Problem with a Riemann Problem
\begin_inset CommandInset label
@@ -1021,11 +1132,21 @@ nolink "false"
\end_inset
the velocity of cars for different densities.
- It can be seen that the velocity decreases linearly for an increasing number of cars,
+ The velocity decreases linearly for an increasing number of cars,
connecting this problem to the traffic flow model.
+ The initial distribution shows a traffic flow on the area
+\begin_inset Formula $x\in[0,1]$
+\end_inset
+
+,
+ while the remaining street is empty.
\end_layout
-\begin_layout Standard
+\begin_layout Remark
+Connecting the flux function to the initial traffic flow problem.
+\end_layout
+
+\begin_layout Remark
The flux has its maximum for a density of
\begin_inset Formula $u=0.5$
\end_inset
@@ -1036,21 +1157,21 @@ The flux has its maximum for a density of
\end_inset
,
- the highest flux is reached for a balance of number of cars and velocity of those.
+ the highest flux is reached for a balance between the number of cars and velocity of those.
For a very small number of cars (
\begin_inset Formula $u\to0$
\end_inset
),
- the velocity of those cars is the highest,
+ those cars can dry very fast,
but the total number of cars passing the street is very small.
On the other hand,
for a very high number of cars (
\begin_inset Formula $u\to1$
\end_inset
-) the velocity of the cars goes close to zero,
- while many cars are on the street.
+) the cars drive very slow,
+ but many cars are on the street.
Therefore,
to reach the highest number of cars going from place A to place B,
the distance from one car to the next car is mandatory.
@@ -1058,11 +1179,7 @@ The flux has its maximum for a density of
decreasing the total number of cars on the street,
nor should it be too small,
decreasing the velocity of those cars on the street.
-
-\end_layout
-
-\begin_layout Standard
-A traffic flow can finally be avoided by maximizing the flow of cars on a street.
+ A traffic flow can finally be avoided by maximizing the flow of cars on a street.
This can be achieved by choosing the number of cars such that the flux function
\begin_inset Formula $f(u)=uV(u)$
\end_inset
@@ -1084,7 +1201,7 @@ name "subsec:AnalyticalSolution_2.3"
\begin_layout Standard
Two solve the Riemann problem analytically,
first note that the flux function is not convex,
- but concave with
+ but concave
\begin_inset Formula
\begin{align}
f(u) & =2u(1-u)\\
@@ -1120,11 +1237,7 @@ nolink "false"
\end_inset
.
-
-\end_layout
-
-\begin_layout Standard
-The Riemann problem at position
+ The Riemann problem at position
\begin_inset Formula $x=0$
\end_inset
@@ -1251,37 +1364,26 @@ w_{r} & ,\xi>f'(w_{r})
& =\begin{cases}
\frac{3}{4} & ,x<1-t\\
\frac{2-\frac{x-1}{t}}{4} & ,1-t\leq x\leq1+2t\\
-0 & ,x>1+2t
+0 & ,1+2t<x
\end{cases}
\end{align}
\end_inset
Combining both Riemann problems again,
- gives the complete analytical solution
+ yields the complete analytical solution
\begin_inset Formula
\begin{equation}
u(x,t)=\begin{cases}
0 & ,x\leq\frac{1}{2}t\\
\frac{3}{4} & ,\frac{1}{2}t<x<1-t\\
\frac{2-\frac{x-1}{t}}{4} & ,1-t\leq x\leq1+2t\\
-0 & ,x>1+2t
+0 & ,1+2t<x
\end{cases}
\end{equation}
\end_inset
-
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-When is it valid?
- And when not?
-\end_layout
-
-\end_inset
-
However,
this solution is only valid for a certain time-period
\begin_inset Formula $t\in[0,T^{c}]$
@@ -1292,7 +1394,6 @@ However,
\end_inset
is the time in which the Rarefaction wave and the shock wave are at the same position.
- Checking the time when the position of the shock wave is at the left side of the Rarefaction wave yields
\begin_inset Formula
\begin{align}
\frac{1}{2}T^{c} & =1-T^{c}\\
@@ -1373,7 +1474,7 @@ status open
\begin_inset Caption Standard
\begin_layout Plain Layout
-Solution at
+Solution at breaking time
\begin_inset Formula $t=T^{c}$
\end_inset
@@ -1382,7 +1483,7 @@ Solution at
\begin_inset Formula $t=T^{c}$
\end_inset
- is visualized for
+ are visualized for
\begin_inset Formula $u$
\end_inset
@@ -1391,6 +1492,7 @@ Solution at
\end_inset
(right).
+ At the breaking time a new problem is formed.
\begin_inset CommandInset label
LatexCommand label
@@ -1449,7 +1551,7 @@ Solution for
\end_inset
.
- The numerical solution (left) and analytical solution (right) show an almost perfect match.
+ The numerical solution (left) and analytical solution (center) show an almost perfect match.
This can be verified by the absolute error (right).
\begin_inset CommandInset label
LatexCommand label
@@ -1529,7 +1631,7 @@ nolink "false"
verifying the correct calculation of the intersection time.
On the right,
- the initial flux and and flux at the intersection time is visualized.
+ the initial flux and the flux at the intersection time are visualized.
Figure
\begin_inset CommandInset ref
LatexCommand ref
@@ -1546,12 +1648,12 @@ nolink "false"
\end_layout
\begin_layout Standard
-Relating the solutions to the physical setup of a traffic flow yields several results
+Relating the solutions to the physical setup of a traffic flow yields
\end_layout
\begin_layout Enumerate
If a traffic flow occurs on the center of a road,
- with no cars before and after the traffic flow,
+ with an empty road elsewhere,
this traffic flow will dissolve from itself
\end_layout
@@ -1575,6 +1677,13 @@ The maximum flux is neither reached for the highest nor the smallest number dens
\begin_layout Subsection
A Second Traffic Flow Simulation with Complex Initial Data
+\begin_inset CommandInset label
+LatexCommand label
+name "subsec:A-Second-Traffic-Flow-Problem"
+
+\end_inset
+
+
\end_layout
\begin_layout Standard
@@ -1628,18 +1737,23 @@ Both distributions are similar on a major part of the domain
,
where the density of cars increases stepwise,
introducing a traffic flow with stepwise increasing significance.
- The number of cars for distribution A decreases on
+ The number of cars for distribution A increases on
\begin_inset Formula $x\in(2.5,3.5]$
\end_inset
- indicating a dissolution of the traffic flow,
- while the significance further increases for distribution B.
+ indicating a further significance of the traffic flow,
+ while it dissolves for initial data B.
+ Initial data B also allows the initial flux to take its maximum on
+\begin_inset Formula $x\in(2.5,3.5]$
+\end_inset
+
+.
\end_layout
\begin_layout Standard
The traffic flow model is again satisfied,
- as the velocity decreases for increasing densities of car (see Section
+ as the velocity decreases for increasing densities of cars (see Section
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Introduction"
@@ -1701,6 +1815,11 @@ Solution for
The traffic flow is not dissolved and its significance is even increased.
At the final time,
the maximum number of cars is reached on the whole street.
+ The initial flux has its maximum on the interior of the domain and a higher value on the inflow than the outflow,
+ leading to a bigger the traffic flow.
+ This flux behavior is conserved over time,
+ which prevents the traffic flow to be dissolved.
+
\begin_inset CommandInset label
LatexCommand label
name "fig:TrafficFlow_A"
@@ -1762,6 +1881,10 @@ Solution for
The rarefaction wave at the right-most Riemann problem reduces the density of cars,
resulting in a free flow of cars,
again.
+ The initial flux function takes its maximum on the outflow.
+ Therefore,
+ more cars can leave the traffic flow than enter.
+ This behavior is conserved over time and allows the traffic flow to dissolve from itself.
\begin_inset CommandInset label
LatexCommand label
name "fig:TrafficFlow_B"
@@ -1831,8 +1954,7 @@ nolink "false"
.
Notice that the traffic flow is successfully dissolved for initial distribution B,
- while the traffic flow's front moves away from the traffic flow for initial distribution B,
- even increasing the significance of the traffic flow.
+ while its significance is even increased for distribution A.
\end_layout
\begin_layout Standard
@@ -1842,12 +1964,12 @@ The flux function is concave (
) and the initial distribution shows several Riemann problems,
at each step.
- For each Riemann problem in initial data A,
-
+ Each Riemann problem in initial data A results in a shock solution,
+ as
\begin_inset Formula $u_{l}<u_{r}$
\end_inset
- indicating a shock solution.
+.
This shock moves towards the left as the shock speed
\begin_inset Formula $s=\frac{f(u_{l})-f(u_{r})}{u_{l}-u_{r}}<0$
\end_inset
@@ -1888,12 +2010,13 @@ Note that the flux function maximizes for
\end_inset
the flux reaches its highest value at the initial distribution and decreases after.
- This results in the increasing significance of the traffic jam.
+ This results in the increasing significance of the traffic jam,
+ as more cars enter the traffic flow than leave it.
Problem B,
on the other hand,
is initialized in a way that the flux of cars is maximized at the end of the traffic flow.
This results in a step-wise dissolution from the traffic jam,
- as the cars in the end of the traffic jam will drive faster compared to those in the beginning and the center of the traffic jam.
+ as more cars leave the traffic jam than enter it.
\begin_inset Note Note
status open
diff --git a/Project2/LyX/TheoryAndMethods.lyx b/Project2/LyX/TheoryAndMethods.lyx
index 833a9e6db8563570dc7421f0c438b4b738dd26ea..e0d067ea6c998f9d7a786d968d6b20e6db337750 100644
--- a/Project2/LyX/TheoryAndMethods.lyx
+++ b/Project2/LyX/TheoryAndMethods.lyx
@@ -127,19 +127,20 @@ and reformulate the problem to
\begin_inset Formula
\begin{equation}
\begin{cases}
-u_{t}+f'(u)u_{x}\\
+u_{t}+f'(u)u_{x}=0\\
u(x,t=0)=u_{0}(x)
\end{cases}\label{eq:gProblem_General}
\end{equation}
\end_inset
-As a first step,
+To find an analytical solution to this problem,
the time derivative of
\begin_inset Formula $u(x(t),t)$
\end_inset
- will be calculated
+ will be calculated,
+ in the first step.
\begin_inset Formula
\begin{equation}
\frac{d}{dt}u(x(t),t)=u_{t}\cancel{\frac{dt}{dt}}+u_{x}\frac{dx}{dt}=u_{t}+u_{x}f'(u)\overset{!}{=}0\label{eq:u_General}
@@ -147,7 +148,7 @@ As a first step,
\end_inset
-from this it can be concluded that
+It can be concluded that
\begin_inset Formula $u(x(t),t)$
\end_inset
@@ -188,7 +189,7 @@ concludes,
\begin_inset Formula
\begin{align}
dx & =f'\left(u_{0}(x_{0})\right)dt\\
-\Leftrightarrow\int_{x(t=0)}^{x(t)}1dx & =\int_{0}^{1}f'\left(u_{0}(x_{0})\right)dt=f'\left(u_{0}(x_{0})\right)t\\
+\Leftrightarrow\int_{x(t=0)}^{x(t)}1dx & =\int_{t=0}^{t}f'\left(u_{0}(x_{0})\right)dt=f'\left(u_{0}(x_{0})\right)t\\
\Rightarrow x(t) & =x_{0}+f'\left(u_{0}(x_{0})\right)t
\end{align}
@@ -231,7 +232,7 @@ A natural arising question is,
.
However,
- the PDE is only fulfilled under the constraint
+ the PDE is only satisfied under the constraint
\begin_inset Formula $1+u_{0}'(x-f'(u(x,t))t)+f''(u(x,t))\neq0$
\end_inset
@@ -247,14 +248,18 @@ literal "false"
\end_layout
\begin_layout Proof
-\begin_inset Formula $1+u_{0}'(x-f'(u(x,t))t)+f''(u(x,t))\neq0$
+\begin_inset Formula $u(x,t)$
\end_inset
- has to be fulfilled to transform
-\begin_inset Formula $u(x,t)$
+ is a solution to the general conservation law
+\begin_inset Formula $u_{t}+f(u)_{x}=0$
\end_inset
- into a solution
+ for
+\begin_inset Formula $1+u_{0}'(x-f'(u(x,t))t)+f''(u(x,t))\neq0$
+\end_inset
+
+
\end_layout
\begin_layout Proof
@@ -264,6 +269,21 @@ First,
\end_inset
.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+Check the derivatives,
+ especially
+\begin_inset Formula $u_{t}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
\end_layout
\begin_layout Proof
@@ -293,7 +313,7 @@ Inserting the expressions for
\begin{align}
u_{t}+f(u)_{x} & =-u_{0}'(v)\left(\cancel{f'(u)}+tf''(u)u_{t}\right)+u_{0}'(v)\left(f\cancel{'(u)}-tf''(u)f(u)_{x}\right)\\
& =-u_{0}'(v)tf''(u)\left(u_{t}+f(u)u_{x}\right)\\
-\Leftrightarrow\left(u_{t}+f(u)_{x}\right) & \left(1+u_{0}'(v)t+f''(u)\right)=0
+\Leftrightarrow\left(u_{t}+f(u)_{x}\right)\left(1+u_{0}'(v)t+f''(u)\right) & =0
\end{align}
\end_inset
@@ -362,11 +382,11 @@ u(x,t) & =u_{0}\left(\underbrace{x-f'(u(x,t)t}_{v(x,t)}\right)
\end_inset
-and calculate the spatial derivative (
+and calculate the spatial derivative
\begin_inset Formula $u_{x}$
\end_inset
-) of this
+ 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)\\
@@ -405,7 +425,7 @@ Rarefaction waves
\end_layout
\begin_layout Standard
-Two different classes of waves can be identifies,
+Two different classes of waves can be identified,
based on the characteristics of the original PDE problem;
rarefaction waves and compression waves.
\end_layout
@@ -414,7 +434,7 @@ Two different classes of waves can be identifies,
Rarefaction waves,
on the one hand,
occur when characteristics diverge and yield smooth and continuous solutions to the PDE.
- The shape of the rarefaction wave is determined by the flux function.
+ The shape of rarefaction waves is determined by the flux function.
Compression waves,
on the other hand,
occur when characteristics converge to each other and will cross.
@@ -646,8 +666,8 @@ end{tikzpicture}
\begin_layout Plain Layout
Rarefaction and Compression waves.
- Rarefaction waves occur for diverging characteristics,
- while compression waves arise for converging characteristics at the point of intersection a discontinuity solution will form.
+ Rarefaction waves occur for diverging characteristics and yield a continuous solution.
+ Compression waves arise for converging characteristics and form a discontinuity at the point of intersection.
\end_layout
\end_inset
@@ -678,13 +698,11 @@ Consider the PDE
\end_inset
.
- The Rankine-Hugoniot condition yields an,
- explicit,
- expression for the speed of a shock (
+ The Rankine-Hugoniot condition yields an explicit expression for the speed of a shock
\begin_inset Formula $s$
\end_inset
-),
+,
given a shock occurs (see
\begin_inset CommandInset citation
LatexCommand cite
@@ -714,7 +732,7 @@ with
\end_layout
\begin_layout Proof
-The Rankine-Hugoniot Condition
+A shock travels with velocity
\begin_inset Formula $s=\frac{f(u_{l})-f(u_{r})}{u_{l}-u_{r}}$
\end_inset
@@ -723,20 +741,12 @@ The Rankine-Hugoniot Condition
\begin_layout Proof
Consider the following assumptions:
-\end_layout
-
-\begin_layout Proof
-The solution to the scalar conservation problem
+ 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
+ yields a shock.
+ The shock moves with a positive velocity along a path and is represented by
\begin_inset Formula $u_{l}(t)$
\end_inset
@@ -744,10 +754,8 @@ The shock,
\begin_inset Formula $u_{r}(t)$
\end_inset
-
-\end_layout
-
-\begin_layout Proof
+.
+
\begin_inset Formula $u_{l}$
\end_inset
@@ -755,11 +763,8 @@ The shock,
\begin_inset Formula $u_{r}$
\end_inset
- are locally constant
-\end_layout
-
-\begin_layout Proof
-The shock speed is represented by
+ are locally constant.
+ The shock speed is represented by
\begin_inset Formula $s=\frac{\Delta x}{\Delta t}$
\end_inset
@@ -771,7 +776,11 @@ 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.
+ around the shock and integrate the PDE around the shock (the window
+\begin_inset Formula $R$
+\end_inset
+
+) over space and time
\end_layout
\begin_layout Proof
@@ -1032,15 +1041,7 @@ Schematic Figure of Rankine-Hugoniot condition window for a convex flux function
\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\\
@@ -1073,7 +1074,7 @@ Now,
\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}
+f(u(x_{1}))=f(u_{l}) & f(u(x_{1}+\Delta x))=f(u_{r})
\end{array}
\end{equation}
@@ -1082,7 +1083,7 @@ f(u(x_{1}))=f(u_{r}) & f(u(x_{1}+\Delta x))=u_{r}
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\\
+0 & =\int_{x_{1}}^{x_{1}+\Delta x}\underbrace{(u_{l}-u_{r})}_{\text{constant}}dx+\int_{t_{1}}^{t_{1}+\Delta t}\underbrace{\left(f(u_{r})-f(u_{l})\right)}_{\text{constant}}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\label{eq:RH}
\end{align}
@@ -1275,7 +1276,7 @@ Schematic visualization of a Riemann problem.
\end_layout
\begin_layout Standard
-For the Riemann problem there can be found two different classes of solutions (see
+There are two classes of solutions for the Riemann problem (see
\begin_inset CommandInset citation
LatexCommand cite
key "Intro_ConsLaw_2021"
@@ -1310,14 +1311,15 @@ Shock Solution:
\end_inset
by a discontinuous solution,
- where the discontinuity moves with the speed s,
+ where the discontinuity moves with speed s,
derived by the Rankine-Hugoniot condition
\end_layout
\begin_layout Standard
-In the following assume the flux function to be convex.
+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.
+ but will produce slightly different results.
\end_layout
\begin_layout Subsubsection
@@ -1335,8 +1337,7 @@ A continuous similarity solution occurs under the condition
\end_inset
,
- where two different characteristics,
- which will not cross.
+ where two different characteristics will not cross.
\begin_inset Formula
\[
x=f'(u_{l})t\qquad x=f'(u_{r})t
@@ -1364,7 +1365,7 @@ In between these two characteristics,
\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})\\
+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}
@@ -1431,7 +1432,7 @@ Where
Note that this inverse exists,
as the flux function is convex and therefore the second derivative is positive.
Now,
- insert the solution
+ insert the equation
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:v_x/t"
@@ -1442,7 +1443,7 @@ nolink "false"
\end_inset
- into the general solution
+ into the equation
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:SimilaritySolutionAbstract"
@@ -1479,7 +1480,7 @@ Contrary to the similarity solution,
\end_inset
.
- For the shock solution two characteristics cross,
+ In this scenario two characteristics cross,
yielding a discontinuous solution with a jump.
This jump moves with the shock speed
\begin_inset Formula $s$
@@ -1503,7 +1504,7 @@ nolink "false"
\begin{equation}
u(x,t)=\begin{cases}
u_{l} & ,\frac{x}{t}\leq s\\
-u_{r} & ,x<\frac{x}{t}
+u_{r} & ,s<\frac{x}{t}
\end{cases}\qquad\text{with: }s=\frac{f(u_{l})-f(u_{r})}{u_{l}-u_{r}}
\end{equation}
@@ -1527,7 +1528,7 @@ Weak solutions can simplify mathematical problems by introducing some test funct
\end_inset
.
- Weak solutions have two important properties
+ Weak solutions have two important properties.
\end_layout
\begin_layout Enumerate
@@ -1544,7 +1545,7 @@ No Derivatives:
\begin_inset Formula $f(u)$
\end_inset
- as they are moved to the test function
+ as they are moved to the test function.
\end_layout
\begin_layout Enumerate
@@ -1554,7 +1555,7 @@ 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
+ Weak solutions can even include more then one solution for a given problem.
\end_layout
\begin_layout Standard
@@ -1591,7 +1592,7 @@ The weak form then reads:
\end_inset
-Note that in the last step integration by parts and the property that the test function vanishes on the boundaries was utilized.
+Note that in the last step integration by parts and the property that the test function vanishes on the boundaries were utilized.
Any function
\begin_inset Formula $u(x,t)$
\end_inset
@@ -1647,7 +1648,7 @@ f'(u_{l})>s>f'(u_{r})
\end_inset
.
- Conclude from this
+ Conclude from this for a convex flux function
\end_layout
\begin_layout Enumerate
diff --git a/Project2/src/Problem_2_3.ipynb b/Project2/src/Problem_2_3.ipynb
index f9791baa5e374130495168c0b3a02348fd887266..532edcb30711b82e806466580d518f16140e31e4 100644
--- a/Project2/src/Problem_2_3.ipynb
+++ b/Project2/src/Problem_2_3.ipynb
@@ -2,7 +2,7 @@
"cells": [
{
"cell_type": "code",
- "execution_count": 37,
+ "execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
@@ -12,7 +12,7 @@
},
{
"cell_type": "code",
- "execution_count": 38,
+ "execution_count": 3,
"metadata": {},
"outputs": [
{
@@ -43,7 +43,7 @@
},
{
"cell_type": "code",
- "execution_count": 39,
+ "execution_count": 4,
"metadata": {},
"outputs": [
{
@@ -89,7 +89,7 @@
},
{
"cell_type": "code",
- "execution_count": 40,
+ "execution_count": 5,
"metadata": {},
"outputs": [
{
@@ -131,7 +131,7 @@
},
{
"cell_type": "code",
- "execution_count": 41,
+ "execution_count": 6,
"metadata": {},
"outputs": [
{
@@ -165,7 +165,7 @@
},
{
"cell_type": "code",
- "execution_count": 42,
+ "execution_count": 7,
"metadata": {},
"outputs": [
{
@@ -174,7 +174,7 @@
"0.375"
]
},
- "execution_count": 42,
+ "execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
@@ -192,7 +192,7 @@
},
{
"cell_type": "code",
- "execution_count": 43,
+ "execution_count": 8,
"metadata": {},
"outputs": [],
"source": [
@@ -202,7 +202,7 @@
},
{
"cell_type": "code",
- "execution_count": 44,
+ "execution_count": 9,
"metadata": {},
"outputs": [],
"source": [
@@ -221,14 +221,14 @@
},
{
"cell_type": "code",
- "execution_count": 45,
+ "execution_count": 10,
"metadata": {},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
- "/var/folders/v_/5q1gkdc53z34pdsfnpkx2t340000gn/T/ipykernel_20214/3866153160.py:17: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n",
+ "/var/folders/v_/5q1gkdc53z34pdsfnpkx2t340000gn/T/ipykernel_49601/3866153160.py:17: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n",
" fig.show()\n"
]
},
@@ -265,7 +265,7 @@
},
{
"cell_type": "code",
- "execution_count": 46,
+ "execution_count": 11,
"metadata": {},
"outputs": [
{
@@ -274,7 +274,7 @@
"(<Figure size 640x480 with 1 Axes>, <Axes: xlabel='x', ylabel='f(u)'>)"
]
},
- "execution_count": 46,
+ "execution_count": 11,
"metadata": {},
"output_type": "execute_result"
},
@@ -297,7 +297,7 @@
},
{
"cell_type": "code",
- "execution_count": 47,
+ "execution_count": 12,
"metadata": {},
"outputs": [
{
@@ -324,7 +324,7 @@
},
{
"cell_type": "code",
- "execution_count": 48,
+ "execution_count": 13,
"metadata": {},
"outputs": [],
"source": [
@@ -351,7 +351,7 @@
},
{
"cell_type": "code",
- "execution_count": 49,
+ "execution_count": 14,
"metadata": {},
"outputs": [
{
@@ -365,7 +365,7 @@
"name": "stderr",
"output_type": "stream",
"text": [
- "/var/folders/v_/5q1gkdc53z34pdsfnpkx2t340000gn/T/ipykernel_20214/1589535761.py:8: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n",
+ "/var/folders/v_/5q1gkdc53z34pdsfnpkx2t340000gn/T/ipykernel_49601/1589535761.py:8: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n",
" fig.show()\n"
]
},
@@ -393,16 +393,16 @@
},
{
"cell_type": "code",
- "execution_count": 50,
+ "execution_count": 15,
"metadata": {},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
- "/var/folders/v_/5q1gkdc53z34pdsfnpkx2t340000gn/T/ipykernel_20214/1239769646.py:12: RuntimeWarning: divide by zero encountered in divide\n",
+ "/var/folders/v_/5q1gkdc53z34pdsfnpkx2t340000gn/T/ipykernel_49601/1239769646.py:12: RuntimeWarning: divide by zero encountered in divide\n",
" (2 - (x-1)/t)/4,\n",
- "/var/folders/v_/5q1gkdc53z34pdsfnpkx2t340000gn/T/ipykernel_20214/346561436.py:3: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n",
+ "/var/folders/v_/5q1gkdc53z34pdsfnpkx2t340000gn/T/ipykernel_49601/346561436.py:3: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n",
" fig.show()\n"
]
},
@@ -425,20 +425,20 @@
},
{
"cell_type": "code",
- "execution_count": 51,
+ "execution_count": 16,
"metadata": {},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
- "/var/folders/v_/5q1gkdc53z34pdsfnpkx2t340000gn/T/ipykernel_20214/1239769646.py:12: RuntimeWarning: divide by zero encountered in divide\n",
+ "/var/folders/v_/5q1gkdc53z34pdsfnpkx2t340000gn/T/ipykernel_49601/1239769646.py:12: RuntimeWarning: divide by zero encountered in divide\n",
" (2 - (x-1)/t)/4,\n",
"/Users/janhabscheid/Documents/git/ddm/Project2/src/SystemGeneric.py:358: RuntimeWarning: divide by zero encountered in divide\n",
" rel_error = abs_error / np.abs(u_analytical)\n",
"/Users/janhabscheid/Documents/git/ddm/Project2/src/SystemGeneric.py:358: RuntimeWarning: invalid value encountered in divide\n",
" rel_error = abs_error / np.abs(u_analytical)\n",
- "/var/folders/v_/5q1gkdc53z34pdsfnpkx2t340000gn/T/ipykernel_20214/71401304.py:3: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n",
+ "/var/folders/v_/5q1gkdc53z34pdsfnpkx2t340000gn/T/ipykernel_49601/71401304.py:3: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n",
" fig.show()\n"
]
},
@@ -477,13 +477,15 @@
},
{
"cell_type": "code",
- "execution_count": 52,
+ "execution_count": 17,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
+ "Nx: [ 10 100 1000 10000]\n",
+ "dx: [0.7 0.07 0.007 0.0007]\n",
"CFL Condition: True, CFL number: 0.00013392857142857144\n",
"CFL Condition: True, CFL number: 0.001339285714285714\n",
"CFL Condition: True, CFL number: 0.013392857142857142\n",
@@ -494,7 +496,7 @@
"name": "stderr",
"output_type": "stream",
"text": [
- "/var/folders/v_/5q1gkdc53z34pdsfnpkx2t340000gn/T/ipykernel_20214/1324504847.py:40: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n",
+ "/var/folders/v_/5q1gkdc53z34pdsfnpkx2t340000gn/T/ipykernel_49601/3459604413.py:43: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n",
" fig.show()\n"
]
},
@@ -544,6 +546,9 @@
"# dx = np.array([10e-3, 10e-2, 10e-1, 10e-0])\n",
"x0, x1 = -2, 5\n",
"dx = (x1 - x0) / Nx\n",
+ "\n",
+ "print(f'Nx: {Nx}')\n",
+ "print(f'dx: {dx}')\n",
"# Nx = (x1 - x0) / dx\n",
"\n",
"NTime = 1_000\n",
@@ -588,7 +593,7 @@
},
{
"cell_type": "code",
- "execution_count": 53,
+ "execution_count": 18,
"metadata": {},
"outputs": [
{
@@ -599,13 +604,13 @@
"<>:16: SyntaxWarning: invalid escape sequence '\\D'\n",
"<>:7: SyntaxWarning: invalid escape sequence '\\D'\n",
"<>:16: SyntaxWarning: invalid escape sequence '\\D'\n",
- "/var/folders/v_/5q1gkdc53z34pdsfnpkx2t340000gn/T/ipykernel_20214/2350136093.py:7: SyntaxWarning: invalid escape sequence '\\D'\n",
+ "/var/folders/v_/5q1gkdc53z34pdsfnpkx2t340000gn/T/ipykernel_49601/2350136093.py:7: SyntaxWarning: invalid escape sequence '\\D'\n",
" axs.set_xlabel('$\\Delta x$')\n",
- "/var/folders/v_/5q1gkdc53z34pdsfnpkx2t340000gn/T/ipykernel_20214/2350136093.py:16: SyntaxWarning: invalid escape sequence '\\D'\n",
+ "/var/folders/v_/5q1gkdc53z34pdsfnpkx2t340000gn/T/ipykernel_49601/2350136093.py:16: SyntaxWarning: invalid escape sequence '\\D'\n",
" axs.set_xlabel('$\\Delta x$')\n",
- "/var/folders/v_/5q1gkdc53z34pdsfnpkx2t340000gn/T/ipykernel_20214/2350136093.py:12: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n",
+ "/var/folders/v_/5q1gkdc53z34pdsfnpkx2t340000gn/T/ipykernel_49601/2350136093.py:12: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n",
" fig.show()\n",
- "/var/folders/v_/5q1gkdc53z34pdsfnpkx2t340000gn/T/ipykernel_20214/2350136093.py:21: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n",
+ "/var/folders/v_/5q1gkdc53z34pdsfnpkx2t340000gn/T/ipykernel_49601/2350136093.py:21: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n",
" fig.show()\n"
]
},
@@ -663,7 +668,7 @@
},
{
"cell_type": "code",
- "execution_count": 54,
+ "execution_count": 19,
"metadata": {},
"outputs": [
{
@@ -677,11 +682,11 @@
"name": "stderr",
"output_type": "stream",
"text": [
- "/var/folders/v_/5q1gkdc53z34pdsfnpkx2t340000gn/T/ipykernel_20214/2362550528.py:2: RuntimeWarning: overflow encountered in multiply\n",
+ "/var/folders/v_/5q1gkdc53z34pdsfnpkx2t340000gn/T/ipykernel_49601/2362550528.py:2: RuntimeWarning: overflow encountered in multiply\n",
" f = lambda u: 2 * u * (1 - u)\n",
"/Users/janhabscheid/Documents/git/ddm/Project2/src/SystemGeneric.py:203: RuntimeWarning: invalid value encountered in subtract\n",
" 1 / self.dx * (upwind_n_1 - upwind_n_2)\n",
- "/var/folders/v_/5q1gkdc53z34pdsfnpkx2t340000gn/T/ipykernel_20214/521724299.py:35: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n",
+ "/var/folders/v_/5q1gkdc53z34pdsfnpkx2t340000gn/T/ipykernel_49601/521724299.py:35: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n",
" fig.show()\n"
]
},