# 1-dimensional Euler equations¶

## Shu-Osher problem¶

Solve the one-dimensional compressible Euler equations:

$\begin{split}\rho_t + (\rho u)_x & = 0 \\ (\rho u)_t + (\rho u^2 + p)_x & = 0 \\ E_t + (u (E + p) )_x & = 0.\end{split}$

The initial condition corresponds to the Shu-Osher problem in which a shock wave impacts a sinusoidally-varying density field.

This example also demonstrates:

• how to use an arbitrary Runge-Kutta method by simply providing the Butcher coefficients of the method.
• How to use a total fluctuation solver in SharpClaw
• How to use characteristic decomposition with an evec() routine in SharpClaw

## Source:¶

#!/usr/bin/env python
# encoding: utf-8
r"""
Shu-Osher problem
====================

Solve the one-dimensional compressible Euler equations:

.. math::
\rho_t + (\rho u)_x & = 0 \\
(\rho u)_t + (\rho u^2 + p)_x & = 0 \\
E_t + (u (E + p) )_x & = 0.

The initial condition corresponds to the Shu-Osher problem
in which a shock wave impacts a sinusoidally-varying density field.

This example also demonstrates:

- how to use an arbitrary Runge-Kutta method by simply providing the
Butcher coefficients of the method.
- How to use a total fluctuation solver in SharpClaw
- How to use characteristic decomposition with an evec() routine in SharpClaw
"""

from __future__ import absolute_import
from __future__ import print_function
import numpy as np
from clawpack import riemann
from clawpack.riemann.euler_with_efix_1D_constants import density, momentum, energy, num_eqn

gamma = 1.4  # Ratio of specific heats

# Coefficients of Runge-Kutta method
a = np.array([[0., 0., 0., 0., 0., 0., 0.],
[.3772689153313680, 0., 0., 0., 0., 0., 0.],
[.3772689153313680, .3772689153313680, 0., 0., 0., 0., 0.],
[.2429952205373960, .2429952205373960, .2429952205373960, 0., 0., 0., 0.],
[.1535890676951260, .1535890676951260, .1535890676951260, .2384589328462900, 0., 0., 0.]])
b = np.array([.206734020864804, .206734020864804, .117097251841844, .181802560120140, .287632146308408])
c = np.array([0., .3772689153313680, .7545378306627360, .7289856616121880, .6992261359316680])

def setup(use_petsc=False,iplot=False,htmlplot=False,outdir='./_output',solver_type='sharpclaw',
kernel_language='Fortran',use_char_decomp=False,tfluct_solver=True):

if use_petsc:
import clawpack.petclaw as pyclaw
else:
from clawpack import pyclaw

if kernel_language =='Python':
rs = riemann.euler_1D_py.euler_roe_1D
elif kernel_language =='Fortran':
rs = riemann.euler_with_efix_1D

if solver_type=='sharpclaw':
solver = pyclaw.SharpClawSolver1D(rs)
solver.time_integrator = 'RK'
solver.a, solver.b, solver.c = a, b, c
solver.cfl_desired = 0.6
solver.cfl_max = 0.7
if use_char_decomp:
try:
from . import sharpclaw1               # Import custom Fortran code
solver.fmod = sharpclaw1
solver.tfluct_solver = tfluct_solver     # Use total fluctuation solver for efficiency
if solver.tfluct_solver:
try:
from . import euler_tfluct
solver.tfluct = euler_tfluct
except ImportError:
import logging
logger = logging.getLogger()
logger.error('Unable to load tfluct solver, did you run make?')
print('Unable to load tfluct solver, did you run make?')
raise
except ImportError:
import logging
logger = logging.getLogger()
logger.error('Unable to load sharpclaw1 solver, did you run make?')
print('Unable to load sharpclaw1 solver, did you run make?')
pass
solver.lim_type = 2             # WENO reconstruction
solver.char_decomp = 2          # characteristic-wise reconstruction
else:
solver = pyclaw.ClawSolver1D(rs)

solver.kernel_language = kernel_language

solver.bc_lower[0]=pyclaw.BC.extrap
solver.bc_upper[0]=pyclaw.BC.extrap

mx = 400;
x = pyclaw.Dimension(-5.0,5.0,mx,name='x')
domain = pyclaw.Domain([x])
state = pyclaw.State(domain,num_eqn)

state.problem_data['gamma']= gamma

if kernel_language =='Python':
state.problem_data['efix'] = False

xc = state.grid.p_centers[0]
epsilon = 0.2
velocity = (xc<-4.)*2.629369
pressure = (xc<-4.)*10.33333 + (xc>=-4.)*1.

state.q[density ,:] = (xc<-4.)*3.857143 + (xc>=-4.)*(1+epsilon*np.sin(5*xc))
state.q[momentum,:] = velocity * state.q[density,:]
state.q[energy  ,:] = pressure/(gamma - 1.) + 0.5 * state.q[density,:] * velocity**2

claw = pyclaw.Controller()
claw.tfinal = 1.8
claw.solution = pyclaw.Solution(state,domain)
claw.solver = solver
claw.num_output_times = 10
claw.outdir = outdir
claw.setplot = setplot
claw.keep_copy = True

return claw

#--------------------------
def setplot(plotdata):
#--------------------------
"""
Specify what is to be plotted at each frame.
Input:  plotdata, an instance of visclaw.data.ClawPlotData.
Output: a modified version of plotdata.
"""
plotdata.clearfigures()  # clear any old figures,axes,items data

# Figure for density
plotfigure = plotdata.new_plotfigure(name='', figno=0)

plotaxes = plotfigure.new_plotaxes()
plotaxes.axescmd = 'subplot(211)'
plotaxes.title = 'Density'
plotaxes.xlimits = (-5.,5.)

plotitem = plotaxes.new_plotitem(plot_type='1d')
plotitem.plot_var = density
plotitem.kwargs = {'linewidth':3}

plotaxes = plotfigure.new_plotaxes()
plotaxes.title = 'Energy'
plotaxes.axescmd = 'subplot(212)'

plotitem = plotaxes.new_plotitem(plot_type='1d')
plotitem.plot_var = energy
plotitem.kwargs = {'linewidth':3}
plotaxes.xlimits = (-5.,5.)

return plotdata

if __name__=="__main__":
from clawpack.pyclaw.util import run_app_from_main
output = run_app_from_main(setup,setplot)