Lagrangian vs pseudo-Lagrangian vs Eulerian

[SpaceNerdz]

I'm reading up a series of papers on hydrodynamical simulations for galaxies and cosmology. They keep mentioning things like "Lagrangian" or "pseudo-lagrangian" or "Eulerian". I tried looking it up on the internet, but the answers are either too broad and could mean a huge number of things in physics, or it does not even show up on Google searches.
Can someone please explain what these three terms mean, and how they are similar to one another, how they differ from one another, and why are they usually being talked about in the same breath ?

 

[jedishrfu]

While I don't have a definitive answer here, I'm throwing some mud at the wall in the hopes that it helps:

I believe these are three different schemes used to solve differential equations in fluid flow problems.

https://en.wikipedia.org/wiki/Lagrangian_and_Eulerian_specification_of_the_flow_field

First there's two possible frames of reference to use in studying fluid flow:
- Eulerian frame of reference where you watch the fluid flow by
- Lagrangian frame of reference where you travel with a differential amount of fluid

Why you choose one frame over the other may have to do with what you're trying to understand about the fluid.

Numerical Computing and ODE Solvers

Why you'd choose one ODE solver over another has to do with whether the system is periodic or is simply increasing/decreasing over time as the Eulerian solvers tend to add/subtract a small amount of error that that over time appears as energy flowing into or out o the system meaning it will become less and less accurate. Sometimes you choose the ODE solver because you need the speed and not the accuracy which favors use of an Eulerian variation over Runge-Kutta.

As an example, if you were to use an Eulerian ODE solver to simulate the Earth orbiting the Sun you might notice the orbit getting larger and larger or smaller and smaller ie spinning off into space or crashing into the Sun whereas using a fancier method such as Runge-Kutta the orbit would vary but would be stable over a long period of time any the error introduced cancels out any earlier error.

Miscellaneous References

You might have to do some further research to understand more on the methods and so I added these papers to help with that as they aren't adequately discussed online. I checked my Intro to Computer Simulations book but couldn't find any direct references to the pseudo-Lagrangian.

The pseudo Lagrangian may in fact be a semi-Lagrangian scheme but I'm not sure:

https://en.wikipedia.org/wiki/Semi-Lagrangian_scheme

Here's a paper that uses Eulerian and pseudo-Lagrangian method to analyze some ocean surface waves:

https://personal.egr.uri.edu/grilli/SG-ISOPE06.pdf

and another using both methods in a galactic simulation:

https://www.slac.stanford.edu/cgi-wrap/getdoc/slac-pub-15677.pdf

and pseudo Lagrangian used in a biochemical ocean study:

https://agupubs.onlinelibrary.wiley.com/doi/full/10.1002/2015JC010898All Things Euler

and from WIkipedia on Euler there's several possible choices:

Euler's ordinary differential equations[edit]

• Euler rotation equations, a set of first-order ODEs concerning the rotations of a rigid body.
• Euler–Cauchy equation, a linear equidimensional second-order ODEs with variable coefficients. Its second-order version can emerge from Laplace equation in polar coordinates.
• Euler–Bernoulli beam equation, a fourth-order ODE concerning the elasticity of structural beams.
• Euler–Lagrange equation, a second-order ODE emerging from minimization problems in calculus of variations.

Euler's partial differential equations[edit]

• Euler conservation equations, a set of quasilinear first-order hyperbolic equations used in fluid dynamics for inviscid flows. In the (Froude) limit of no external field, they are conservation equations.
• Euler–Tricomi equation – a second-order PDE emerging from Euler conservation equations.
• Euler–Poisson–Darboux equation, a second-order PDE playing important role in solving the wave equation.