Demo of Van Leer convection term in FiPy showing how it preserves the shock quite well

vanleer.py

from fipy import CellVariable, Grid1D, TransientTerm
from fipy import MatplotlibViewer, VanLeerConvectionTerm
from fipy import FaceVariable, ImplicitSourceTerm

length = 1.
nx = 100
dx = length/nx
mesh = Grid1D(dx=dx, nx=nx)

TG = CellVariable(mesh = mesh, hasOld = True, name = 'TG')
T_G_init = 20.0 + 273.15
TG.setValue(T_G_init)
T_G_in = 100.0 + 273.15
TG.constrain(T_G_in, mesh.facesLeft)

m_G_dot = 0.1
rho_G = 1.1  
v = m_G_dot / rho_G
dt_CFL = dx/v

coeff = FaceVariable(mesh=mesh, value=[v], rank=1)
coeff.setValue([0], where=mesh.facesRight)
coeffRight = FaceVariable(mesh=mesh, value=[0.], rank=1)
coeffRight.setValue([v], where=mesh.facesRight)

eqn = TransientTerm() \
  + VanLeerConvectionTerm(coeff) \
  + ImplicitSourceTerm(coeffRight.divergence) == 0

iterTime = length/v
iterSteps = iterTime/dt_CFL
iterSteps = int(iterSteps)

viewer = MatplotlibViewer(vars=(TG),datamin=273.15, datamax=400)

for t in xrange(iterSteps + 20):
    res = 1e+10
    TG.updateOld()
    while res > 1e-10:
        res = eqn.sweep(var=TG, dt=dt_CFL)
    viewer.plot()

I updated the Gist this issue was that the right hand boundary condition again wasn't working correctly with the constraint. I am not sure what the issue is, but you can work around it by using an ImplicitSourceTerm to imitate an outflow condition on the right hand side.