1-dimensional advection

One-dimensional advection

Solve the linear advection equation:

\[q_t + u q_x = 0.\]

Here q is the density of some conserved quantity and u is the velocity.

The initial condition is a Gaussian and the boundary conditions are periodic. The final solution is identical to the initial data because the wave has crossed the domain exactly once.

Output:

../../_images/pyclaw_examples_advection_1d__plots_advection_1d_frame0000fig1.png ../../_images/pyclaw_examples_advection_1d__plots_advection_1d_frame0004fig1.png ../../_images/pyclaw_examples_advection_1d__plots_advection_1d_frame0010fig1.png

Source:

#!/usr/bin/env python
# encoding: utf-8

r"""
One-dimensional advection
=========================

Solve the linear advection equation:

.. math:: 
    q_t + u q_x = 0.

Here q is the density of some conserved quantity and u is the velocity.

The initial condition is a Gaussian and the boundary conditions are periodic.
The final solution is identical to the initial data because the wave has
crossed the domain exactly once.
"""
from __future__ import absolute_import
import numpy as np
from clawpack import riemann

def setup(nx=100, kernel_language='Python', use_petsc=False, solver_type='classic', weno_order=5, 
          time_integrator='SSP104', outdir='./_output'):

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

    if kernel_language == 'Fortran':
        riemann_solver = riemann.advection_1D
    elif kernel_language == 'Python':
        riemann_solver = riemann.advection_1D_py.advection_1D
            
    if solver_type=='classic':
        solver = pyclaw.ClawSolver1D(riemann_solver)
    elif solver_type=='sharpclaw':
        solver = pyclaw.SharpClawSolver1D(riemann_solver)
        solver.weno_order = weno_order
        solver.time_integrator = time_integrator
        if time_integrator == 'SSPLMMk3':
            solver.lmm_steps = 5
            solver.check_lmm_cond = True
    else: raise Exception('Unrecognized value of solver_type.')

    solver.kernel_language = kernel_language

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

    x = pyclaw.Dimension(0.0,1.0,nx,name='x')
    domain = pyclaw.Domain(x)
    state = pyclaw.State(domain,solver.num_eqn)

    state.problem_data['u'] = 1.  # Advection velocity

    # Initial data
    xc = state.grid.x.centers
    beta = 100; gamma = 0; x0 = 0.75
    state.q[0,:] = np.exp(-beta * (xc-x0)**2) * np.cos(gamma * (xc - x0))

    claw = pyclaw.Controller()
    claw.keep_copy = True
    claw.solution = pyclaw.Solution(state,domain)
    claw.solver = solver

    if outdir is not None:
        claw.outdir = outdir
    else:
        claw.output_format = None

    claw.tfinal =1.0
    claw.setplot = setplot

    return claw

def setplot(plotdata):
    """ 
    Plot solution using VisClaw.
    """ 
    plotdata.clearfigures()  # clear any old figures,axes,items data

    plotfigure = plotdata.new_plotfigure(name='q', figno=1)

    # Set up for axes in this figure:
    plotaxes = plotfigure.new_plotaxes()
    plotaxes.ylimits = [-.2,1.0]
    plotaxes.title = 'q'

    # Set up for item on these axes:
    plotitem = plotaxes.new_plotitem(plot_type='1d_plot')
    plotitem.plot_var = 0
    plotitem.plotstyle = '-o'
    plotitem.color = 'b'
    plotitem.kwargs = {'linewidth':2,'markersize':5}
    
    return plotdata

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