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
9,46331347
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
9,46331347
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
9,46331347
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
9,46331347
asked Dec 31 '18 at 16:01
Dirac_DeltaDirac_Delta
242
edited Jan 2 at 16:41
Harry49
9,46331347
asked Dec 31 '18 at 16:01
Dirac_DeltaDirac_Delta
242
edited Jan 2 at 16:41
Harry49
9,46331347
edited Jan 2 at 16:41
Harry49
9,46331347
edited Jan 2 at 16:41
Harry49
9,46331347
9,46331347
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
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$begingroup$
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