Traffic flow with Dirac-$delta$ source (on ramp)
$begingroup$
I have been trying to solve the traffic flow equation with a singular source ($D>0$ large):
$$
rho_t + f(rho)_x = Ddelta(x)
$$
with the flux $f(rho)=rho(1-rho)$ and the initial data $rho(x,0)=0.4$.
I understand that the jump condition for small $D$ is $f(rho_r)-f(rho_l)=D$, where $rho_l=0.4$. This gives me a real value for $rho_r$. But when $D$ is large (eg:0.012), it does not give me a real value for $rho_r$, so there should be something non trivial happening, like a shock (which makes sense physically too). However, I am not able to work this out explicitly. I tried to incorporate a shock term to the jump condition, but that gave me two unknowns in one equation.
Does anyone have any ideas on how I could proceed?
pde dirac-delta hyperbolic-equations
edited Jan 2 at 16:41
Harry49
7,67431344
asked Dec 31 '18 at 16:01
Dirac_DeltaDirac_Delta
242
$endgroup$
|
show 3 more comments
$begingroup$
I have been trying to solve the traffic flow equation with a singular source ($D>0$ large):
$$
rho_t + f(rho)_x = Ddelta(x)
$$
with the flux $f(rho)=rho(1-rho)$ and the initial data $rho(x,0)=0.4$.
I understand that the jump condition for small $D$ is $f(rho_r)-f(rho_l)=D$, where $rho_l=0.4$. This gives me a real value for $rho_r$. But when $D$ is large (eg:0.012), it does not give me a real value for $rho_r$, so there should be something non trivial happening, like a shock (which makes sense physically too). However, I am not able to work this out explicitly. I tried to incorporate a shock term to the jump condition, but that gave me two unknowns in one equation.
Does anyone have any ideas on how I could proceed?
pde dirac-delta hyperbolic-equations
edited Jan 2 at 16:41
Harry49
7,67431344
asked Dec 31 '18 at 16:01
Dirac_DeltaDirac_Delta
242
$endgroup$
$begingroup$
Just thinking out loud, here. Looks like $rho_r$ goes imaginary when $D>1/100,$ based on your value for $rho_l.$ Can you edit your question to include what you had in mind for a shock term? Maybe we can work around the lack of information problem you mentioned.
$endgroup$
– Adrian Keister
Dec 31 '18 at 16:24
$begingroup$
For $xnot=0,$ your pde reduces down to $rho_t+rho_x-2rhorho_x=0.$ The IC, of course, makes the trivial solution not work. Again, still just thinking out loud, here.
$endgroup$
– Adrian Keister
Dec 31 '18 at 16:29
$begingroup$
This is a hyperbolic conservation law, which doesn't necessarily have a unique solution. The setup you have is a generalised Riemann problem, which may include shocks and rarefaction fans.
$endgroup$
– Eddy
Dec 31 '18 at 16:35
1
$begingroup$
However, the shock condition and jump condition give you 2 equations, and there are 2 unknowns, so you should only have finitely many solutions.
$endgroup$
– Eddy
Dec 31 '18 at 16:37
$begingroup$
The constant solution $rho=2/5$ satisfies the pde and IC when $xnot=0$.
$endgroup$
– Adrian Keister
Dec 31 '18 at 16:42
|
show 3 more comments
$begingroup$
I have been trying to solve the traffic flow equation with a singular source ($D>0$ large):
$$
rho_t + f(rho)_x = Ddelta(x)
$$
with the flux $f(rho)=rho(1-rho)$ and the initial data $rho(x,0)=0.4$.
I understand that the jump condition for small $D$ is $f(rho_r)-f(rho_l)=D$, where $rho_l=0.4$. This gives me a real value for $rho_r$. But when $D$ is large (eg:0.012), it does not give me a real value for $rho_r$, so there should be something non trivial happening, like a shock (which makes sense physically too). However, I am not able to work this out explicitly. I tried to incorporate a shock term to the jump condition, but that gave me two unknowns in one equation.
Does anyone have any ideas on how I could proceed?
pde dirac-delta hyperbolic-equations
edited Jan 2 at 16:41
Harry49
7,67431344
asked Dec 31 '18 at 16:01
Dirac_DeltaDirac_Delta
242
$endgroup$
I have been trying to solve the traffic flow equation with a singular source ($D>0$ large):
$$
rho_t + f(rho)_x = Ddelta(x)
$$
with the flux $f(rho)=rho(1-rho)$ and the initial data $rho(x,0)=0.4$.
I understand that the jump condition for small $D$ is $f(rho_r)-f(rho_l)=D$, where $rho_l=0.4$. This gives me a real value for $rho_r$. But when $D$ is large (eg:0.012), it does not give me a real value for $rho_r$, so there should be something non trivial happening, like a shock (which makes sense physically too). However, I am not able to work this out explicitly. I tried to incorporate a shock term to the jump condition, but that gave me two unknowns in one equation.
Does anyone have any ideas on how I could proceed?
pde dirac-delta hyperbolic-equations
pde dirac-delta hyperbolic-equations
edited Jan 2 at 16:41
Harry49
7,67431344
asked Dec 31 '18 at 16:01
Dirac_DeltaDirac_Delta
242
edited Jan 2 at 16:41
Harry49
7,67431344
asked Dec 31 '18 at 16:01
Dirac_DeltaDirac_Delta
242
edited Jan 2 at 16:41
Harry49
7,67431344
edited Jan 2 at 16:41
Harry49
7,67431344
edited Jan 2 at 16:41
Harry49
7,67431344
7,67431344
asked Dec 31 '18 at 16:01
Dirac_DeltaDirac_Delta
242
asked Dec 31 '18 at 16:01
Dirac_DeltaDirac_Delta
242
asked Dec 31 '18 at 16:01
Dirac_DeltaDirac_Delta
242
242
$begingroup$
Just thinking out loud, here. Looks like $rho_r$ goes imaginary when $D>1/100,$ based on your value for $rho_l.$ Can you edit your question to include what you had in mind for a shock term? Maybe we can work around the lack of information problem you mentioned.
$endgroup$
– Adrian Keister
Dec 31 '18 at 16:24
$begingroup$
For $xnot=0,$ your pde reduces down to $rho_t+rho_x-2rhorho_x=0.$ The IC, of course, makes the trivial solution not work. Again, still just thinking out loud, here.
$endgroup$
– Adrian Keister
Dec 31 '18 at 16:29
$begingroup$
This is a hyperbolic conservation law, which doesn't necessarily have a unique solution. The setup you have is a generalised Riemann problem, which may include shocks and rarefaction fans.
$endgroup$
– Eddy
Dec 31 '18 at 16:35
1
$begingroup$
However, the shock condition and jump condition give you 2 equations, and there are 2 unknowns, so you should only have finitely many solutions.
$endgroup$
– Eddy
Dec 31 '18 at 16:37
$begingroup$
The constant solution $rho=2/5$ satisfies the pde and IC when $xnot=0$.
$endgroup$
– Adrian Keister
Dec 31 '18 at 16:42
|
show 3 more comments
$begingroup$
Just thinking out loud, here. Looks like $rho_r$ goes imaginary when $D>1/100,$ based on your value for $rho_l.$ Can you edit your question to include what you had in mind for a shock term? Maybe we can work around the lack of information problem you mentioned.
$endgroup$
– Adrian Keister
Dec 31 '18 at 16:24
$begingroup$
For $xnot=0,$ your pde reduces down to $rho_t+rho_x-2rhorho_x=0.$ The IC, of course, makes the trivial solution not work. Again, still just thinking out loud, here.
$endgroup$
– Adrian Keister
Dec 31 '18 at 16:29
$begingroup$
This is a hyperbolic conservation law, which doesn't necessarily have a unique solution. The setup you have is a generalised Riemann problem, which may include shocks and rarefaction fans.
$endgroup$
– Eddy
Dec 31 '18 at 16:35
1
$begingroup$
However, the shock condition and jump condition give you 2 equations, and there are 2 unknowns, so you should only have finitely many solutions.
$endgroup$
– Eddy
Dec 31 '18 at 16:37
$begingroup$
The constant solution $rho=2/5$ satisfies the pde and IC when $xnot=0$.
$endgroup$
– Adrian Keister
Dec 31 '18 at 16:42
$begingroup$
Just thinking out loud, here. Looks like $rho_r$ goes imaginary when $D>1/100,$ based on your value for $rho_l.$ Can you edit your question to include what you had in mind for a shock term? Maybe we can work around the lack of information problem you mentioned.
$endgroup$
– Adrian Keister
Dec 31 '18 at 16:24
$begingroup$
Just thinking out loud, here. Looks like $rho_r$ goes imaginary when $D>1/100,$ based on your value for $rho_l.$ Can you edit your question to include what you had in mind for a shock term? Maybe we can work around the lack of information problem you mentioned.
$endgroup$
– Adrian Keister
Dec 31 '18 at 16:24
$begingroup$
For $xnot=0,$ your pde reduces down to $rho_t+rho_x-2rhorho_x=0.$ The IC, of course, makes the trivial solution not work. Again, still just thinking out loud, here.
$endgroup$
– Adrian Keister
Dec 31 '18 at 16:29
$begingroup$
For $xnot=0,$ your pde reduces down to $rho_t+rho_x-2rhorho_x=0.$ The IC, of course, makes the trivial solution not work. Again, still just thinking out loud, here.
$endgroup$
– Adrian Keister
Dec 31 '18 at 16:29
$begingroup$
This is a hyperbolic conservation law, which doesn't necessarily have a unique solution. The setup you have is a generalised Riemann problem, which may include shocks and rarefaction fans.
$endgroup$
– Eddy
Dec 31 '18 at 16:35
$begingroup$
This is a hyperbolic conservation law, which doesn't necessarily have a unique solution. The setup you have is a generalised Riemann problem, which may include shocks and rarefaction fans.
$endgroup$
– Eddy
Dec 31 '18 at 16:35
1
1
$begingroup$
However, the shock condition and jump condition give you 2 equations, and there are 2 unknowns, so you should only have finitely many solutions.
$endgroup$
– Eddy
Dec 31 '18 at 16:37
$begingroup$
However, the shock condition and jump condition give you 2 equations, and there are 2 unknowns, so you should only have finitely many solutions.
$endgroup$
– Eddy
Dec 31 '18 at 16:37
$begingroup$
The constant solution $rho=2/5$ satisfies the pde and IC when $xnot=0$.
$endgroup$
– Adrian Keister
Dec 31 '18 at 16:42
$begingroup$
The constant solution $rho=2/5$ satisfies the pde and IC when $xnot=0$.
$endgroup$
– Adrian Keister
Dec 31 '18 at 16:42
|
show 3 more comments
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3057837%2ftraffic-flow-with-dirac-delta-source-on-ramp%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3057837%2ftraffic-flow-with-dirac-delta-source-on-ramp%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
Just thinking out loud, here. Looks like $rho_r$ goes imaginary when $D>1/100,$ based on your value for $rho_l.$ Can you edit your question to include what you had in mind for a shock term? Maybe we can work around the lack of information problem you mentioned.
$endgroup$
– Adrian Keister
Dec 31 '18 at 16:24
$begingroup$
For $xnot=0,$ your pde reduces down to $rho_t+rho_x-2rhorho_x=0.$ The IC, of course, makes the trivial solution not work. Again, still just thinking out loud, here.
$endgroup$
– Adrian Keister
Dec 31 '18 at 16:29
$begingroup$
This is a hyperbolic conservation law, which doesn't necessarily have a unique solution. The setup you have is a generalised Riemann problem, which may include shocks and rarefaction fans.
$endgroup$
– Eddy
Dec 31 '18 at 16:35
1
$begingroup$
However, the shock condition and jump condition give you 2 equations, and there are 2 unknowns, so you should only have finitely many solutions.
$endgroup$
– Eddy
Dec 31 '18 at 16:37
$begingroup$
The constant solution $rho=2/5$ satisfies the pde and IC when $xnot=0$.
$endgroup$
– Adrian Keister
Dec 31 '18 at 16:42