I modified this example (http://www.karlin.mff.cuni.cz/~hron/fenics-tutorial/discontinuous_galerkin/doc.html). I replaced the initial condition with a uniform scalar quantity and a convection field uniform that goes downwards (0.0, -2.0). I also removed the boundary conditions. Now I am interested in having the scalar quantity accumulate in the bottom boundary because of the convection field. How can I do this? I figured that I need to enforce zero flux at the bottom boundary. I don't know much about DG fluxes, which method is the one implemented here? SUPG? Thanks
import time
import os
import math
from dolfin import *
# get file name
fileName = os.path.splitext(__file__)[0]
parameters['form_compiler']['cpp_optimize'] = True
parameters['form_compiler']['optimize'] = True
parameters["ghost_mode"] = "shared_facet"
# Parameters
Pe = Constant(1e10)
t_end = 10
dt = 0.1
# Create mesh and define function space
mesh = RectangleMesh(0, 0, 1, 1, 40, 40, 'crossed')
# Define function spaces
V = FunctionSpace(mesh, "DG", 1)
ic= Constant(0.0)
b = Expression(("0.0","-2.0"), domain=mesh)
bc=DirichletBC(V,Constant(0.0),DomainBoundary(), method="geometric")
# Define unknown and test function(s)
v = TestFunction(V)
u = TrialFunction(V)
u0 = Function(V)
u0 = interpolate(ic,V )
# STABILIZATION
h = CellSize(mesh)
n = FacetNormal(mesh)
alpha = Constant(1e0)
theta = Constant(1.0)
# ( dot(v, n) + |dot(v, n)| )/2.0
bn = (dot(b, n) + abs(dot(b, n)))/2.0
def a(u,v) :
# Bilinear form
a_int = dot(grad(v), (1.0/Pe)*grad(u) - b*u)*dx
a_fac = (1.0/Pe)*(alpha/avg(h))*dot(jump(u, n), jump(v, n))*dS \
- (1.0/Pe)*dot(avg(grad(u)), jump(v, n))*dS \
- (1.0/Pe)*dot(jump(u, n), avg(grad(v)))*dS
a_vel = dot(jump(v), bn('+')*u('+') - bn('-')*u('-') )*dS + dot(v, bn*u)*ds
a = a_int + a_fac + a_vel
return a
# Define variational forms
a0=a(u0,v)
a1=a(u,v)
A = (1/dt)*inner(u, v)*dx - (1/dt)*inner(u0,v)*dx + theta*a1 + (1-theta)*a0
F = A
# Create files for storing results
file = File("u.pvd")
u = Function(V)
ffc_options = {"optimize": True, "quadrature_degree": 8}
problem = LinearVariationalProblem(lhs(F),rhs(F), u, [bc], form_compiler_parameters=ffc_options)
solver = LinearVariationalSolver(problem)
u.assign(u0)
u.rename("u", "u")
# Time-stepping
t = 0.0
file << u
while t < t_end:
print "t =", t, "end t=", t_end
# Compute
solver.solve()
plot(u)
# Save to file
file << u
# Move to next time step
u0.assign(u)
t += dt
