A Nonlinear Time Dependent Problem

Now we can set up our PDE model As an example we will study the Forchheimer problem [Kie15] which is a scalar, nonlinear parabolic equation \begin{equation} \partial_tu - \nabla\cdot K(\nabla u)\nabla u = f \end{equation} where the diffusion tensor is given by \begin{equation} K(\nabla u) = \frac{2}{1+\sqrt{1+4|\nabla u|}} \end{equation} and \(f=f(x,t)\) is some time dependent forcing term. On the boundary we prescribe Neumann boundary as before and initial conditions \(u=u_0\).

We will solve this problem in variational form and using Crank Nicholson in time \begin{equation} \begin{split} \int_{\Omega} \frac{u^{n+1}-u^n}{\Delta t} \varphi + \frac{1}{2}K(\nabla u^{n+1}) \nabla u^{n+1} \cdot \nabla \varphi \ + \frac{1}{2}K(\nabla u^n) \nabla u^n \cdot \nabla \varphi v\ dx \\ - \int_{\Omega} \frac{1}{2}(f(x,t^n)+f(x,t^n+\Delta t) \varphi\ dx - \int_{\partial \Omega} \frac{1}{2}(g(x,t^n)+g(x,t^n+\Delta t)) v\ ds = 0. \end{split} \end{equation} on a domain \(\Omega=[0,1]^2\). We choose \(f,g\) so that the exact solution is given by \begin{align*} u(x,t) = e^{-2t}\left(\frac{1}{2}(x^2 + y^2) - \frac{1}{3}(x^3 - y^3)\right) + 1 \end{align*}

import matplotlib
matplotlib.rc( 'image', cmap='jet' )
import sys, io

from dune.grid import structuredGrid as leafGridView
from dune.fem.space import lagrange as solutionSpace
from dune.fem.scheme import galerkin as solutionScheme
from dune.fem.function import gridFunction, cppFunction, integrate, uflFunction
from dune.ufl import DirichletBC, Constant
from ufl import TestFunction, TrialFunction, SpatialCoordinate, FacetNormal, \
                dx, ds, div, grad, dot, inner, sqrt, exp, sin

gridView = leafGridView([0, 0], [1, 1], [10, 10])
space = solutionSpace(gridView, order=2)

x = SpatialCoordinate(space)
initial = 1/2*(x[0]**2+x[1]**2) - 1/3*(x[0]**3 - x[1]**3) + 1
exact   = lambda t: exp(-2*t)*(initial - 1) + 1

u_h   = space.interpolate(initial, name='u_h')
u_h_n = u_h.copy(name="previous")

u = TrialFunction(space)
v = TestFunction(space)
dt = Constant(0, name="dt")    # time step
t  = Constant(0, name="t")     # current time

abs_du = lambda u: sqrt(inner(grad(u), grad(u)))
K = lambda u: 2/(1 + sqrt(1 + 4*abs_du(u)))
a = ( dot((u - u_h_n)/dt, v) \
    + 0.5*dot(K(u)*grad(u), grad(v)) \
    + 0.5*dot(K(u_h_n)*grad(u_h_n), grad(v)) ) * dx

f = lambda s: -2*exp(-2*s)*(initial - 1) - div( K(exact(s))*grad(exact(s)) )
g = lambda s: K(exact(s))*grad(exact(s))
n = FacetNormal(space)
b = 0.5*(f(t)+f(t+dt))*v*dx + 0.5*dot(g(t)+g(t+dt),n)*v*ds

With the model described as a ufl form, we can construct a scheme class that provides the solve method which we can use to evolve the solution from one time step to the next:

scheme = solutionScheme(a == b, solver='cg')

Optionally, we may want to increase the default quadrature orders which are ‘2 * space.order’ for element integrals and ‘2 * space.order + 1’ for surface integrals. Depending on the data this might not be enough. Then we simply set the integration orders by hand like in the following example, by calling the method setQuadratureOrders(interiorOrder, surfaceOrder).

#scheme.setQuadratureOrders( 2*space.order, 2*space.order+1 )

Since we have forced the system towards a given solution, we can compute the discretization error. First we define ufl expressions for the \(L^2\) and \(H^1\) norms and will use those to compute the experimental order of convergence of the scheme by computing the time evolution on different grid levels.

endTime    = 0.25
exact_end  = exact(endTime)
l2error = uflFunction(gridView, name="l2error", order=u_h.space.order, ufl=dot(u_h - exact_end, u_h - exact_end))
h1error = uflFunction(gridView, name="h1error", order=u_h.space.order, ufl=dot(grad(u_h - exact_end), grad(u_h - exact_end)))

Now we evolve the solution from time \(t=0\) to \(t=T\) in a loop. Since the PDE model has time dependent coefficient (through the forcing term), we need to update the t constant used to define the model before each step. A second constant we used to define the model was dt which defines the time step. We keep this constant and set it to \(0.002\) at the beginning of the simulation. This value could be changed in each time step:

scheme.model.dt = 0.002
time = 0
while time < (endTime - 1e-6):
    scheme.model.t = time
    time += scheme.model.dt

errors = [sqrt(e) for e in integrate(gridView, [l2error,h1error], order=5)]
print('grid size:', gridView.size(0))
print('\t | u_h - u | =', '{:0.5e}'.format(errors[0]))
print('\t | grad(uh - u) | =', '{:0.5e}'.format(errors[1]))
gridView.writeVTK('forchheimer', pointdata={'u': u_h, 'l2error': l2error, 'h1error': h1error})
grid size: 100
         | u_h - u | = 8.26004e-06
         | grad(uh - u) | = 6.39420e-04

This page was generated from the notebook dune-fempy_nb.ipynb and is part of the tutorial for the dune-fem python bindings DOI