[hunt_mat]

I am trying to solve a simple set of equations numerically but using Lagrangian co-ordinates. In Eulerian co-ordinates they are:

rho_t+(u*rho)_x=0
u_t+u*u_x=D*u_xx

In the included notes I have included the derivation for everything. I have two questions:
1) Does h change all the time? Or can I keep it at the original configuration?
2) How would I compute the length of my new domain?

[FMDenaro]

You have to write your system along the characteristics, this way




Drho/Dt=-rho*du/dx


Du/Dt=D*d^2u/dx^2


and integrate along the curve dx/dt=u(x(t),t)


D/Dt being the lagrangian derivative.

[hunt_mat]

So you can only solve equations in Lagrangian co-ordinates using the method of characteristics and not the way that I suggested in my note?

[FMDenaro]

Originally Posted by hunt_mat

So you can only solve equations in Lagrangian co-ordinates using the method of characteristics and not the way that I suggested in my note?

You asked for the lagrangian formulation, that is the equations are written along the path-line.

Note that the viscous Burgers equation is independent from the density. There is an exact solution for it, see Hopf-Cole transformation.

[hunt_mat]

This isn't helping me much in understanding the numerics of Lagrangian co-ordinates. I'm not interested in what clever methods you use to reduce the complexities of the governing equations, these equations are not the be-all and end-all of the equations I wish to study. I'm using these as a starting point to study more complicated equations in lagrangian co-ordinates.

I always thought there was some massive secret in how one did the numerics in Lagrangian co-ordinates, your answers aren't helping.

Let me be clear:
1) Is the method I proposed in the note correct? Can you solve the equations this way?

2) How do you compute the length using Lagrangian co-ordinates?

[FMDenaro]

Originally Posted by hunt_mat

This isn't helping me much in understanding the numerics of Lagrangian co-ordinates. I'm not interested in what clever methods you use to reduce the complexities of the governing equations, these equations are not the be-all and end-all of the equations I wish to study. I'm using these as a starting point to study more complicated equations in lagrangian co-ordinates.

I always thought there was some massive secret in how one did the numerics in Lagrangian co-ordinates, your answers aren't helping.

Let me be clear:
1) Is the method I proposed in the note correct? Can you solve the equations this way?

2) How do you compute the length using Lagrangian co-ordinates?

You asked how to approach the solution of the equation in lagrangian formulation.
You can invent any new mathematical formulation you want (or just discover an old mathematical formulation), this is not my task, and is not what I addressed in terms of lagrangian formulation.

In lagrangian formulation you solve along the path-line the equation. The key is to transform the x-derivative in the RHS along the lagrangian system.
Start from solving the independent Burgers equation on the lagrangian coordinates to understand the task.

Just as hint, you can see how the method works numerically in the Zucrow texbook, the section dedicated to 1d unsteady Euler equations for isothermal flows where you find a similar issue of integrating the RHS along the characteristics.

[hunt_mat]

So method of characteristics then.

Thank you. It's not so useful for me then. I think I'll go back to the more complicated approach, as it can be generalised.

[FMDenaro]

Originally Posted by hunt_mat

So method of characteristics then.

Thank you. It's not so useful for me then. I think I'll go back to the more complicated approach, as it can be generalised.

My idea of any lagrangian method is that you write the total differential

du =dt*( du/dt + dx/dt du/dx)

on the path-line dx/dt=u. Then integrate the equation along the local lagrangian system.

That is independent from the concept of characteristic lines that can be also different curves.

[hunt_mat]

Originally Posted by FMDenaro

My idea of any lagrangian method is that you write the total differential

du =dt*( du/dt + dx/dt du/dx)

on the path-line dx/dt=u. Then integrate the equation along the local lagrangian system.

That is independent from the concept of characteristic lines that can be also different curves.

That seems independent of type of co-ordinates as well.

[FMDenaro]

Originally Posted by hunt_mat

That seems independent of type of co-ordinates as well.

Coordinate dare defined by the path-line x=x(t)

[hunt_mat]

How does that tie in with my co-ordinate h?

[FMDenaro]

Originally Posted by hunt_mat

How does that tie in with my co-ordinate h?

From your definition dh/dx= rho, clearly h is not a lenght. Why you want to use something like density*length as coordinate??

That means you define h(x(t)) by integrating rho up to x(t).

[hunt_mat]

I'm aware it isn't a length. I explained what h was, and how it could be used as a co-ordinate. It's quite common apparently.

[FMDenaro]

Originally Posted by hunt_mat

I'm aware it isn't a length. I explained what h was, and how it could be used as a co-ordinate. It's quite common apparently.

I don't know why you introduced something reminding me to the Stewartson-Dorodnitzjn transformation used in the compressible BL.

[hunt_mat]

It's the amount of mass in the interval considered. It's a common Lagrangian co-ordinate system used in 1D. It's mentioned in LeVeque's book.

[FMDenaro]

Originally Posted by hunt_mat

It's the amount of mass in the interval considered. It's a common Lagrangian co-ordinate system used in 1D. It's mentioned in LeVeque's book.

But why you need of that?? Once solved the Burgers equation, the density equation is independent since du/dx is known.

How do you apply your trasformation into the single Burgers equation if you don’t have tue density variable?

[hunt_mat]

Originally Posted by FMDenaro

But why you need of that?? Once solved the Burgers equation, the density equation is independent since du/dx is known.

How do you apply your trasformation into the single Burgers equation if you don’t have tue density variable?

These equations are a warm-up to more complicated equations which have the same formulation. I explained this explicitly before, that's why I am uninterested in techniques for specific equations. I want to solve some simple equations first, before I go on to the more complicated equations later on.

[FMDenaro]

Originally Posted by hunt_mat

These equations are a warm-up to more complicated equations which have the same formulation. I explained this explicitly before, that's why I am uninterested in techniques for specific equations. I want to solve some simple equations first, before I go on to the more complicated equations later on.

But how do you solve the Burgers equation with your trasformation involving a variable that is not present?
That makes no sense.

[hunt_mat]

Originally Posted by FMDenaro

But how do you solve the Burgers equation with your trasformation involving a variable that is not present?
That makes no sense.

You don't. This co-ordinate is only relevant when you have the conservation of mass equation along with another equation.

[FMDenaro]

Originally Posted by hunt_mat

You don't. This co-ordinate is only relevant when you have the conservation of mass equation along with another equation.

You realize now that you have to write the system with the full momentum equation by adding -(1/rho) dp/dx. Then you can use the iso-entropic relation (if you problem allows such hypothesis) and close the system of two equations.