# 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.

## 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 = pyclaw.BC.periodic
solver.bc_upper = 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)