Euler equations: exact solutions¶

This notebook illustrates some features of the exact solution to the 1-dimensional Euler equations of compressible gas dynamics.

The notebook can be found in the clawpack apps respository:

$CLAW/apps/notebooks/riemann/Riemann_problem_Euler_exact.ipynb
See http://www.clawpack.org/apps.html.

In [1]:

%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
from IPython.display import FileLink

The next cell imports a module containing a function that takes a Riemann problem (left state, right state, and approximate solver), and computes the Riemann solution, as well as functions to plot the solution in various forms.

In [2]:

#from clawpack.riemann import riemann_tools

# the version from clawpack.riemann needs updating, 
# for active development use the version in this repository instead:
import sys
sys.path.insert(0,'..')  
import riemann_tools

We can compute the exact solution to the Riemann problem for the Euler equations.

In [3]:

import Euler_exact_Riemann_solver
FileLink('Euler_exact_Riemann_solver.py')  # Link to examine the exact Riemann solver

Out[3]:

Euler_exact_Riemann_solver.py

In [4]:

gamma = 1.4

in_vars = 'primitive'  # so components of q are (rho,u,p)

q_l = np.array((3.,0.,3.))
q_r = np.array((1.,0.,1.))

states, speeds, riemann_eval = Euler_exact_Riemann_solver.exact_riemann_solution(q_l ,q_r, gamma, 
                                                                in_vars=in_vars, out_vars='primitive')

Note that we specified out_vars = 'primitive' above, so that the states returned are in terms of the primitive variables (density, velocity, pressure). This allows us to confirm that the pressure and velocity are constant across the 2-wave (contact discontinuity) and only the density has a jump discontinuity.

Since the phase space is 3-dimensional, we show two different projections, to density-velocity and to density-pressure:

In [5]:

fig = plt.figure(figsize=(8,3))
ax = plt.subplot(1,2,1)
riemann_tools.plot_phase(states, ax=ax, i_h = 0, i_v = 1, label_h='density', label_v='velocity')
ax = plt.subplot(1,2,2)
riemann_tools.plot_phase(states, ax=ax, i_h = 0, i_v = 2, label_h='density', label_v='pressure')

In both cases we see that the variable on the vertical axis is constant between the two middle states.

We can also see this in the solution plotted below.

In [6]:

riemann_tools.JSAnimate_plot_riemann(states,speeds,riemann_eval)

Out[6]: