Euler Sod Shock Tube Problem
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Dear Johannes,
Thanks for making this public !!
I would like to apply your method to the the astrophysics field
in a context where one has to solve hyperbolic PDEs with shocks
similar, for instance, to the 1D Euler equations (coupled system
of equations).
My question is whether is possible to use your method
in such a type of PDEs.
Thanks,
Roberto
Dear Roberto,
thank you very much for you interest!
Can you provide more information about your PDE (dimensionality, regular/non-regular grid, boundary conditions, ...)?
Thanks,
Johannes
Dear Johannes,
It's a hyperbolic PDE system with three coupled equations in the x (ie. 1 spatial component) and t domain.
The equation is the following:
Forward Problem for 1-D Euler Equations for Compressible Flow: The Sod Problem
In this code we solve for rho, u, p from the following equations:
U_t + AU_x = 0, (x,t) in (0,1)x(0,0.2]
(rho,u,p)_t=0 = (1.0,0.0,1.0) 0 <= x < 0.5
(rho,u,p)_t=0 = (0.125,0.0,0.1) 0.5 <= x <=1
with Dirichlet boundary conditions which take the values of the initial condition at the boundaries
= [ rho ] and A = [ u, rho, 0 ]
[ u ] [ 0, u, 1/rho ]
[ p ] [ 0, gammap, u ]
rho -- Density of the fluid
u -- Velocity of the fluid - x direction
p -- Pressure of the fluid
E -- Total energy of fluid
We relate the pressure and energy by the equation of state of the form
p = (gamma - 1) ( rhoE - 0.5*rho||u||^2)
For this problem we use gamma = 1.4
Thanks !!
Rbt
BTW I attach a solution using numerical methods (pyclaw) and a standard PINN method for t = 0.2 sec. The idea is to see if your method can do better than the PINN in the discontinuities.
Thanks,
Rbt
Dear Roberto,
thank you very much for the detailed information. Yes, this in fact exactly looks like the kind of problem we where trying to tackle.
I assume you will have a non-regular grid?
Instead of having an output (input) of n_timestepsn_x you will now have n_timestepsn_x*4 -> rho, u, p, E. T
Johannes
Hi Johannes,
The spatial grid can be regular between 0 and 1 (ie. taking a fixed number of steps). I guess you mean that, right ?
I will have a look at your code and try to implement it. If would be good if your approach works in this context since in these problems (ie. standard in astrophysics) pinns fail.
Will you be able to give me a hand in the implementation if I have some issue ?
Thanks a lot !!
Rbt
Yes this is what I meant!
Please don't hesitate to contact me, I am happy to assist for the implementation.
I guess we can close this issue and follow-up via email :)
Yes sure.
Thanks a lot !!