1-dimensional shallow water equation

Shallow water flow

Solve the one-dimensional shallow water equations:

\[\begin{split}h_t + (hu)_x & = 0 \\ (hu)_t + (hu^2 + \frac{1}{2}gh^2)_x & = 0.\end{split}\]

Here h is the depth, u is the velocity, and g is the gravitational constant. The default initial condition used here models a dam break.

Output:

Source:

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

r"""
Shallow water flow
==================

Solve the one-dimensional shallow water equations:

.. math::
    h_t + (hu)_x & = 0 \\
    (hu)_t + (hu^2 + \frac{1}{2}gh^2)_x & = 0.

Here h is the depth, u is the velocity, and g is the gravitational constant.
The default initial condition used here models a dam break.
"""

import numpy as np
from clawpack import riemann
from clawpack.riemann.shallow_roe_with_efix_1D_constants import depth, momentum, num_eqn

def setup(use_petsc=False,kernel_language='Fortran',outdir='./_output',solver_type='classic',
          riemann_solver='roe', disable_output=False):

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

    if kernel_language == 'Python':
        if riemann_solver.lower() == 'roe':
            raise Exception('Python Roe solver not implemented.')
        elif riemann_solver.lower() == 'hlle':
            rs = riemann.shallow_1D_py.shallow_hll_1D
    elif kernel_language == 'Fortran':
        if riemann_solver.lower() == 'roe':
            rs = riemann.shallow_roe_with_efix_1D
        elif riemann_solver.lower() == 'hlle':
            rs = riemann.shallow_hlle_1D
 
    if solver_type == 'classic':
        solver = pyclaw.ClawSolver1D(rs)
        solver.limiters = pyclaw.limiters.tvd.vanleer
    elif solver_type == 'sharpclaw':
        solver = pyclaw.SharpClawSolver1D(rs)

    solver.kernel_language = kernel_language

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

    xlower = -5.0
    xupper = 5.0
    mx = 500
    x = pyclaw.Dimension(xlower,xupper,mx,name='x')
    domain = pyclaw.Domain(x)
    state = pyclaw.State(domain,num_eqn)

    # Gravitational constant
    state.problem_data['grav'] = 1.0
    state.problem_data['dry_tolerance'] = 1e-3
    state.problem_data['sea_level'] = 0.0
    
    xc = state.grid.x.centers

    IC='dam-break'
    x0=0.

    if IC=='dam-break':
        hl = 3.
        ul = 0.
        hr = 1.
        ur = 0.
        state.q[depth,:] = hl * (xc <= x0) + hr * (xc > x0)
        state.q[momentum,:] = hl*ul * (xc <= x0) + hr*ur * (xc > x0)
    elif IC=='2-shock':
        hl = 1.
        ul = 1.
        hr = 1.
        ur = -1.
        state.q[depth,:] = hl * (xc <= x0) + hr * (xc > x0)
        state.q[momentum,:] = hl*ul * (xc <= x0) + hr*ur * (xc > x0)
    elif IC=='perturbation':
        eps=0.1
        state.q[depth,:] = 1.0 + eps*np.exp(-(xc-x0)**2/0.5)
        state.q[momentum,:] = 0.

    claw = pyclaw.Controller()
    claw.keep_copy = True
    if disable_output:
        claw.output_format = None
    claw.tfinal = 2.0
    claw.solution = pyclaw.Solution(state,domain)
    claw.solver = solver
    claw.outdir = outdir
    claw.setplot = setplot

    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 depth
    plotfigure = plotdata.new_plotfigure(name='Water height', figno=0)

    # Set up for axes in this figure:
    plotaxes = plotfigure.new_plotaxes()
    plotaxes.xlimits = [-5.0,5.0]
    plotaxes.title = 'Water height'
    plotaxes.axescmd = 'subplot(211)'

    # Set up for item on these axes:
    plotitem = plotaxes.new_plotitem(plot_type='1d')
    plotitem.plot_var = depth
    plotitem.plotstyle = '-'
    plotitem.color = 'b'
    plotitem.kwargs = {'linewidth':3}

    # Figure for momentum[1]
    #plotfigure = plotdata.new_plotfigure(name='Momentum', figno=1)

    # Set up for axes in this figure:
    plotaxes = plotfigure.new_plotaxes()
    plotaxes.axescmd = 'subplot(212)'
    plotaxes.xlimits = [-5.0,5.0]
    plotaxes.title = 'Momentum'

    # Set up for item on these axes:
    plotitem = plotaxes.new_plotitem(plot_type='1d')
    plotitem.plot_var = momentum
    plotitem.plotstyle = '-'
    plotitem.color = 'b'
    plotitem.kwargs = {'linewidth':3}
    
    return plotdata


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