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

Output:

../../_images/pyclaw_examples_euler_1d__plots_shocksine_frame0000fig0.png ../../_images/pyclaw_examples_euler_1d__plots_shocksine_frame0003fig0.png ../../_images/pyclaw_examples_euler_1d__plots_shocksine_frame0010fig0.png

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)