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Cuchulainn
Posts: 62114
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
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### Re: About solving a transport equation

1) You like transforming coords. So change to coords that move with the chars.

Or

2) Constrain dy and dt so that you solve numerically along the chars. That’s what I’ve used in the past.
1) Don't understand that comment. I just used several standard fd schemes directly on Test Case 2.
2) That is possible; however, it is not easy for me and it is difficult to generalise, seems to be the general consensus.

Cuchulainn
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Joined: July 16th, 2004, 7:38 am
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### Re: About solving a transport equation

A good next example is the inviscid (and possibly viscous) Burgers' equation using the methods discussed to date.

Test Case 3

$\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} = 0.$ ... advective form (3A)

or

$\frac{\partial u}{\partial t} + \frac {1}{2}\frac{\partial u^2}{\partial x} = 0.$ .. conservative form (3B)

Eventually, the viscous PDE

$\frac{\partial u}{\partial t} + \frac {1}{2}\frac{\partial u^2}{\partial x} = \nu \frac{\partial^2 u}{\partial x^2.}$  (3C)

https://en.wikipedia.org/wiki/Burgers%27_equation

It would be interesting if Alan and Paul can produce explicit solution for (3A)/(3B)

$u(x,t) = (ax + b)/(at + 1)$

for the initial condition

$u(x,0) = ax + b.$

FaridMoussaoui
Posts: 507
Joined: June 20th, 2008, 10:05 am
Location: Genève, Genf, Ginevra, Geneva

### Re: About solving a transport equation

Take any (intro) book on numerical schemes for conservative laws. You can go with Roe, HLLC or FCT and you are done.

Alan
Posts: 10165
Joined: December 19th, 2001, 4:01 am
Location: California
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### Re: About solving a transport equation

https://en.wikipedia.org/wiki/Burgers%27_equation

It would be interesting if Alan and Paul can produce explicit solution for (3A)/(3B)

$u(x,t) = (ax + b)/(at + 1)$

for the initial condition

$u(x,0) = ax + b.$
I said earlier I don't know much about shocks. But, reading that Wikipedia link, it's trivial to start with their "implicit relation" $u = f(x - u \, t)$, and linear $f$, and then find the given solution for times such that the denominator does not vanish: $t < -1/a$. Presumably from the physics, $a < 0$ is the interesting case.

It's also easy to see that, if u solves the general implicit relation (and f has a derivative), then you get $(u_t + u u_x)(1 + f' t) =0$. So, unless $1 + f' t = 0$, you get a general PDE problem solution. The vanishing of the second factor must lead to Wikipedia's formula for their "breaking time" $t_b$, but needs some thought. Their formula for that does check out for the linear case example.

Cuchulainn
Posts: 62114
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
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### Re: About solving a transport equation

Take any (intro) book on numerical schemes for conservative laws. You can go with Roe, HLLC or FCT and you are done.
That's not the main goal, really. See it more as a benchmark case as it were. We can test many methods on it.
But we want to test Roe et al (mainly Laney) against other schemes. I am more interested in my own discoveries etc, And I am sure Alan and Paul will have fresh viewpoints.

BTW Farid, which methods do you recommend for (3A), (3B)?

Most of the books don't have the detail.
Last edited by Cuchulainn on February 13th, 2020, 4:49 pm, edited 2 times in total.

Paul
Posts: 10771
Joined: July 20th, 2001, 3:28 pm

### Re: About solving a transport equation

I don't have any fresh viewpoints! I'm just regurgitating some basic things I learned 40 years ago!

Cuchulainn
Posts: 62114
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
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### Re: About solving a transport equation

I don't have any fresh viewpoints! I'm just regurgitating some basic things I learned 40 years ago!
Total Recall?

Cuchulainn
Posts: 62114
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
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### Re: About solving a transport equation

https://en.wikipedia.org/wiki/Burgers%27_equation

It would be interesting if Alan and Paul can produce explicit solution for (3A)/(3B)

$u(x,t) = (ax + b)/(at + 1)$

for the initial condition

$u(x,0) = ax + b.$
I said earlier I don't know much about shocks. But, reading that Wikipedia link, it's trivial to start with their "implicit relation" $u = f(x - u \, t)$, and linear $f$, and then find the given solution for times such that the denominator does not vanish: $t < -1/a$. Presumably from the physics, $a < 0$ is the interesting case.

It's also easy to see that, if u solves the general implicit relation (and f has a derivative), then you get $(u_t + u u_x)(1 + f' t) =0$. So, unless $1 + f' t = 0$, you get a general PDE problem solution. The vanishing of the second factor must lead to Wikipedia's formula for their "breaking time" $t_b$, but needs some thought. Their formula for that does check out for the linear case example.
Burgers' etc. is new for me too (Paul and Farid have a 40 year headstart).

This looks like a nice overview.
https://www.iist.ac.in/sites/default/files/people/Burgers_equation_inviscid.pdf

Paul
Posts: 10771
Joined: July 20th, 2001, 3:28 pm

### Re: About solving a transport equation

Cat $\in$ pigeons:

Jump conditions

Weak solutions

Multiply 3(A) by any power of u and write in conservation form

FaridMoussaoui
Posts: 507
Joined: June 20th, 2008, 10:05 am
Location: Genève, Genf, Ginevra, Geneva

### Re: About solving a transport equation

BTW Farid, which methods do you recommend for (3A), (3B)?
use any second-oder Riemann solver.

FaridMoussaoui
Posts: 507
Joined: June 20th, 2008, 10:05 am
Location: Genève, Genf, Ginevra, Geneva

### Re: About solving a transport equation

There is a software out there called CLAWPACK. Under the hood, it is classical fortran routines supplied with python interface.

Have a look to the gallery of test cases.

As usual the installation on Unix-like systems (Linux/OSX) is seamless. They said nothing about Windows OS but if you install Cygwin it should work.
When I switch to windows OS, I use it a lot (cygwin) as I like working with command line.

Cuchulainn
Posts: 62114
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
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### Re: About solving a transport equation

OK, I installed Cygwin and will take that route now.
My main interest is finding out how and why these numerical methods work. Maybe it's called reverse engineering/osmosis.

I am looking at approximate Riemann solvers, In fact, my upwind scheme is similar to Roe's method since it is essentially a linearisation of a quasilinear pde.
BTW Tannehill et al book is a nice overview of some of these methods.

FaridMoussaoui
Posts: 507
Joined: June 20th, 2008, 10:05 am
Location: Genève, Genf, Ginevra, Geneva

### Re: About solving a transport equation

A good book for this kind of solvers is Toro  "Riemann Solvers and Numerical Methods for Fluid Dynamics"

A good intro paper is "Wave Propagation Algorithms for Multidimensional Hyperbolic Systems"

If you can't access freely the paper, I can send you a copy.

Cuchulainn
Posts: 62114
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
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### Re: About solving a transport equation

Thx!

JohnLeM
Posts: 380
Joined: September 16th, 2008, 7:15 pm

### Re: About solving a transport equation

Take any (intro) book on numerical schemes for conservative laws. You can go with Roe, HLLC or FCT and you are done.
That's not the main goal, really. See it more as a benchmark case as it were. We can test many methods on it.
But we want to test Roe et al (mainly Laney) against other schemes. I am more interested in my own discoveries etc, And I am sure Alan and Paul will have fresh viewpoints.

BTW Farid, which methods do you recommend for (3A), (3B)?

Most of the books don't have the detail.
Trying to be useful there, I just mention that we proposed some years ago  in this paper a method to compute explicitly solutions to these equations : more precisely one can compute explicitly (entropic or conservative) multi-dimensional solutions for conservation laws or Hamilton-Jacobi type equations, with convex fluxes (of Burger type) or non convex fluxes (to model multi-phase medium).