Hyperbolic equation in conservative form

TurbJet · February 20, 2018, 13:58

say, for the most common hyperbolic equation

$\partial_{t}u + \partial_{x}[f(u)] = 0$

Let's be specific, say

(it's the LWH traffic model) where the

is a function of u, which is

and the

, u is the density of car flow.

To solve this equation, of course I have to find out the CFL condition, but for the conservative form, the coefficient in front of convective term is 1 which means dt is bounded by 1; however it seems the "wave velocity" should be $\frac{\partial f}{\partial u}$ which aka phase velocity. But this requires to use the product rule of the derivative and it kind of like break the conservative form and it becomes

$\partial_{t}u + v\partial_{x}u + u\partial_{x}v = 0$ ,

and now the velocity used for computing CFL should be v. But when actually solve this equation, should I back to the conservative form, or just stay with the one after product rule?

And a more general question: for conservative form, how should I determine the CFL condition? What if I have a complex f(u) and not sure which one is the velocity I should use for CFL condition? Is it still the phase velocity I mentioned above? or is it something else?

FMDenaro · February 20, 2018, 14:20

When you consider d/dx f(u) = df/du *du/dx you express the quasi-linear form of the conservation law

du/dt + f'(u) du/dx =0

Have a look to the traffic flow equation problem described in the book of Leveque (page 203).

TurbJet · February 20, 2018, 15:57

Quote:

Originally Posted by FMDenaro

When you consider d/dx f(u) = df/du *du/dx you express the quasi-linear form of the conservation law

du/dt + f'(u) du/dx =0

Have a look to the traffic flow equation problem described in the book of Leveque (page 203).

Thx for the recommendation. I ran through the section, what I get is that, we can only determine the time step size by taking the quasi-linear form of hyperbolic equation; but when we are trying to solve the equation itself, we typically go back to the conservative form...

FMDenaro · February 20, 2018, 16:00

Quote:

Originally Posted by TurbJet

Thx for the recommendation. I ran through the section, what I get is that, we can only determine the time step size by taking the quasi-linear form of hyperbolic equation; but when we are trying to solve the equation itself, we typically go back to the conservative form...

The quai linear form is useful to determine the solution in terms of the characteristic lines. Numerically, the conservative form is strongly adviced.

TurbJet · February 20, 2018, 16:03

Quote:

Originally Posted by FMDenaro

The quai linear form is useful to determine the solution in terms of the characteristic lines. Numerically, the conservative form is strongly adviced.

To determine the time step size, use f'(u), right?

FMDenaro · February 20, 2018, 16:26

Quote:

Originally Posted by TurbJet

To determine the time step size, use f'(u), right?

You need to determine the max value of convective velocity

TurbJet · February 20, 2018, 16:31

Quote:

Originally Posted by FMDenaro

You need to determine the max value of convective velocity

this is where I am not sure: to determine the max convective velocity, use f'(u), correct?

praveen · February 20, 2018, 22:51

Quote:

Originally Posted by TurbJet

this is where I am not sure: to determine the max convective velocity, use f'(u), correct?

Yes, f'(u) is the local wave speed.

FMDenaro · February 21, 2018, 04:40

Quote:

Originally Posted by TurbJet

this is where I am not sure: to determine the max convective velocity, use f'(u), correct?

Just consider that the path-lines dx/dx=u does not coincide with the characteristic lines dx/dt=f'(u). That modifies the geometric meaning of the CFL

TurbJet · February 21, 2018, 20:03

Quote:

Originally Posted by praveen

Yes, f'(u) is the local wave speed.

appreciate it!

February 20, 2018, 13:58	Hyperbolic equation in conservative form	#1
TurbJet Senior Member Join Date: Oct 2017 Location: United States Posts: 233 Blog Entries: 1 Rep Power: 10	say, for the most common hyperbolic equation $\partial_{t}u + \partial_{x}[f(u)] = 0$ Let's be specific, say (it's the LWH traffic model) where the is a function of u, which is and the , u is the density of car flow. To solve this equation, of course I have to find out the CFL condition, but for the conservative form, the coefficient in front of convective term is 1 which means dt is bounded by 1; however it seems the "wave velocity" should be $\frac{\partial f}{\partial u}$ which aka phase velocity. But this requires to use the product rule of the derivative and it kind of like break the conservative form and it becomes $\partial_{t}u + v\partial_{x}u + u\partial_{x}v = 0$ , and now the velocity used for computing CFL should be v. But when actually solve this equation, should I back to the conservative form, or just stay with the one after product rule? And a more general question: for conservative form, how should I determine the CFL condition? What if I have a complex f(u) and not sure which one is the velocity I should use for CFL condition? Is it still the phase velocity I mentioned above? or is it something else?

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February 20, 2018, 14:20		#2
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,897 Rep Power: 73	When you consider d/dx f(u) = df/du *du/dx you express the quasi-linear form of the conservation law du/dt + f'(u) du/dx =0 Have a look to the traffic flow equation problem described in the book of Leveque (page 203).