DG advection equation with upwinding

We next consider the advection equation

\[\frac{\partial q}{\partial t} + (\vec{u}\cdot\nabla)q = 0\]

in a domain \(\Omega\), where \(\vec{u}\) is a prescribed vector field, and \(q(\vec{x}, t)\) is an unknown scalar field. The value of \(q\) is known initially:

\[q(\vec{x}, 0) = q_0(\vec{x}),\]

and the value of \(q\) is known for all time on the subset of the boundary \(\Gamma\) in which \(\vec{u}\) is directed towards the interior of the domain:

\[q(\vec{x}, t) = q_\mathrm{in}(\vec{x}, t) \quad \text{on} \ \Gamma_\mathrm{inflow}\]

where \(\Gamma_\mathrm{inflow}\) is defined appropriately.

We will look for a solution \(q\) in a space of discontinuous functions \(V\). A weak form of the continuous equation in each element \(e\) is

\[\int_e \! \phi_e \frac{\partial q}{\partial t} \, \mathrm{d} x + \int_e \! \phi_e (\vec{u}\cdot\nabla)q \, \mathrm{d} x = 0, \qquad \forall \phi_e \in V_e,\]

where we explicitly introduce the subscript \(e\) since the test functions \(\phi_e\) are local to each element. Using integration by parts on the second term, we get

\[\int_e \! \phi_e \frac{\partial q}{\partial t} \, \mathrm{d} x = \int_e \! q \nabla \cdot (\phi_e \vec{u}) \, \mathrm{d} x - \int_{\partial e} \! \phi_e q \vec{u} \cdot \vec{n}_e \, \mathrm{d} S, \qquad \forall \phi_e \in V_e,\]

where \(\vec{n}_e\) is an outward-pointing unit normal.

Since \(q\) is discontinuous, we have to make a choice about how to define \(q\) on facets when we assemble the equations globally. We will use upwinding: we choose the upstream value of \(q\) on facets, with respect to the velocity field \(\vec{u}\). We note that there are three types of facets that we may encounter:

  1. Interior facets. Here, the value of \(q\) from the upstream side, denoted \(\widetilde{q}\), is used.
  2. Inflow boundary facets, where \(\vec{u}\) points towards the interior. Here, the upstream value is the prescribed boundary value \(q_\mathrm{in}\).
  3. Outflow boundary facets, where \(\vec{u}\) points towards the outside. Here, the upstream value is the interior solution value \(q\).

We must now express our problem in terms of integrals over the entire mesh and over the sets of interior and exterior facets. This is done by summing our earlier expression over all elements \(e\). The cell integrals are easy to handle, since \(\sum_e \int_e \cdot \,\mathrm{d}x = \int_\Omega \cdot \,\mathrm{d}x\). The interior facet integrals are more difficult to express, since each facet in the set of interior facets \(\Gamma_\mathrm{int}\) appears twice in the \(\sum_e \int_{\partial e}\). In other words, contributions arise from both of the neighbouring cells.

In Firedrake, the separate quantities in the two cells neighbouring an interior facet are denoted by + and -. These markings are arbitrary – there is no built-in concept of upwinding, for example – and the user is responsible for providing a form that works in all cases. We will give an example shortly. The exterior facet integrals are easier to handle, since each facet in the set of exterior facets \(\Gamma_\mathrm{ext}\) appears exactly once in \(\sum_e \int_{\partial e}\). The full equations are then

\[\int_\Omega \! \phi \frac{\partial q}{\partial t} \, \mathrm{d} x = \int_\Omega \! q \nabla \cdot (\phi \vec{u}) \, \mathrm{d} x - \int_{\Gamma_\mathrm{int}} \! \widetilde{q}(\phi_+ \vec{u} \cdot \vec{n}_+ + \phi_- \vec{u} \cdot \vec{n}_-) \, \mathrm{d} S - \int_{\Gamma_\rlap{\mathrm{ext, inflow}}} \phi q_\mathrm{in} \vec{u} \cdot \vec{n} \, \mathrm{d} s - \int_{\Gamma_\rlap{\mathrm{ext, outflow}}} \phi q \vec{u} \cdot \vec{n} \, \mathrm{d} s \qquad \forall \phi \in V.\]

As a timestepping scheme, we use the three-stage strong-stability-preserving Runge-Kutta (SSPRK) scheme from [SO88]: to discretise \(\frac{\partial q}{\partial t} = \mathcal{L}(q)\), we set

\[\begin{split}q^{(1)} &= q^n + \Delta t \mathcal{L}(q^n)\\ q^{(2)} &= \frac{3}{4}q^n + \frac{1}{4}(q^{(1)} + \Delta t \mathcal{L}(q^{(1)}))\\ q^{n+1} &= \frac{1}{3}q^n + \frac{2}{3}(q^{(2)} + \Delta t \mathcal{L}(q^{(2)}))\\\end{split}\]

In this worked example, we reproduce the classic cosine-bell–cone–slotted-cylinder advection test case of [LeV96]. The domain \(\Omega\) is the unit square \(\Omega = [0,1] \times [0,1]\), and the velocity field corresponds to solid body rotation \(\vec{u} = (0.5 - y, x - 0.5)\). Each side of the domain has a section of inflow and a section of outflow boundary. We therefore perform both the inflow and outflow integrals over the entire boundary, but construct them so that they only contribute in the correct places.

As usual, we start by importing Firedrake. We also import the math library to give us access to the value of pi. We use a 40-by-40 mesh of squares.

from firedrake import *
import math

mesh = UnitSquareMesh(40, 40, quadrilateral=True)

We set up a function space of discontinous bilinear elements for \(q\), and a vector-valued continuous function space for our velocity field.

V = FunctionSpace(mesh, "DQ", 1)
W = VectorFunctionSpace(mesh, "CG", 1)

We set up the initial velocity field using a simple analytic expression.

x, y = SpatialCoordinate(mesh)

velocity = as_vector((0.5 - y, x - 0.5))
u = Function(W).interpolate(velocity)

Now, we set up the cosine-bell–cone–slotted-cylinder initial coniditon. The first four lines declare various parameters relating to the positions of these objects, while the analytic expressions appear in the last three lines.

bell_r0 = 0.15; bell_x0 = 0.25; bell_y0 = 0.5
cone_r0 = 0.15; cone_x0 = 0.5; cone_y0 = 0.25
cyl_r0 = 0.15; cyl_x0 = 0.5; cyl_y0 = 0.75
slot_left = 0.475; slot_right = 0.525; slot_top = 0.85

bell = 0.25*(1+cos(math.pi*min_value(sqrt(pow(x-bell_x0, 2) + pow(y-bell_y0, 2))/bell_r0, 1.0)))
cone = 1.0 - min_value(sqrt(pow(x-cone_x0, 2) + pow(y-cone_y0, 2))/cyl_r0, 1.0)
slot_cyl = conditional(sqrt(pow(x-cyl_x0, 2) + pow(y-cyl_y0, 2)) < cyl_r0,
             conditional(And(And(x > slot_left, x < slot_right), y < slot_top),
               0.0, 1.0), 0.0)

We then declare the inital condition of \(q\) to be the sum of these fields. Furthermore, we add 1 to this, so that the initial field lies between 1 and 2, rather than between 0 and 1. This ensures that we can’t get away with neglecting the inflow boundary condition. We also save the initial state so that we can check the \(L^2\)-norm error at the end.

q = Function(V).interpolate(1.0 + bell + cone + slot_cyl)
q_init = Function(V).assign(q)

We declare the output filename, and write out the initial condition.

outfile = File("DGadv.pvd")
outfile.write(q)

We will run for time \(2\pi\), a full rotation. We take 600 steps, giving a timestep close to the CFL limit. We declare an extra variable dtc; for technical reasons, this means that Firedrake does not have to compile new C code if the user tries different timesteps. Finally, we define the inflow boundary condition, \(q_\mathrm{in}\). In general, this would be a Function, but here we just use a Constant value.

T = 2*math.pi
dt = T/600.0
dtc = Constant(dt)
q_in = Constant(1.0)

Now we declare our variational forms. Solving for \(\Delta q\) at each stage, the explicit timestepping scheme means that the left hand side is just a mass matrix.

dq_trial = TrialFunction(V)
phi = TestFunction(V)
a = phi*dq_trial*dx

The right-hand-side is more interesting. We define n to be the built-in FacetNormal object; a unit normal vector that can be used in integrals over exterior and interior facets. We next define un to be an object which is equal to \(\vec{u}\cdot\vec{n}\) if this is positive, and zero if this is negative. This will be useful in the upwind terms.

n = FacetNormal(mesh)
un = 0.5*(dot(u, n) + abs(dot(u, n)))

We now define our right-hand-side form L1 as \(\Delta t\) times the sum of four integrals.

The first integral is a straightforward cell integral of \(q\nabla\cdot(\phi\vec{u})\). The second integral represents the inflow boundary condition. We only want this to contribute on the inflow part of the boundary, where \(\vec{u}\cdot\vec{n} < 0\) (recall that \(\vec{n}\) is an outward-pointing normal). Where this is true, the condition gives the desired expression \(\phi q_\mathrm{in}\vec{u}\cdot\vec{n}\), otherwise the condition gives zero. The third integral operates in a similar way to give the outflow boundary condition. The last integral represents the integral \(\widetilde{q}(\phi_+ \vec{u} \cdot \vec{n}_+ + \phi_- \vec{u} \cdot \vec{n}_-)\) over interior facets. We could again use a conditional in order to represent the upwind value \(\widetilde{q}\) by the correct choice of \(q_+\) or \(q_-\), depending on the sign of \(\vec{u}\cdot\vec{n_+}\), say. Instead, we make use of the quantity un, which is either \(\vec{u}\cdot\vec{n}\) or zero, in order to avoid writing explicit conditionals. Although it is not obvious at first sight, the expression given in code is equivalent to the desired expression, assuming \(\vec{n}_- = -\vec{n}_+\).

L1 = dtc*(q*div(phi*u)*dx
          - conditional(dot(u, n) < 0, phi*dot(u, n)*q_in, 0.0)*ds
          - conditional(dot(u, n) > 0, phi*dot(u, n)*q, 0.0)*ds
          - (phi('+') - phi('-'))*(un('+')*q('+') - un('-')*q('-'))*dS)

In our Runge-Kutta scheme, the first step uses \(q^n\) to obtain \(q^{(1)}\). We therefore declare similar forms that use \(q^{(1)}\) to obtain \(q^{(2)}\), and \(q^{(2)}\) to obtain \(q^{n+1}\). We make use of UFL’s replace feature to avoid writing out the form repeatedly.

q1 = Function(V); q2 = Function(V)
L2 = replace(L1, {q: q1}); L3 = replace(L1, {q: q2})

We now declare a variable to hold the temporary increments at each stage.

dq = Function(V)

Since we want to perform hundreds of timesteps, ideally we should avoid reassembling the left-hand-side mass matrix each step, as this does not change. We therefore make use of the LinearVariationalProblem and LinearVariationalSolver objects for each of our Runge-Kutta stages. These cache and reuse the assembled left-hand-side matrix. Since the DG mass matrices are block-diagonal, we use the ‘preconditioner’ ILU(0) to solve the linear systems. As a minor technical point, we in fact use an outer block Jacobi preconditioner. This allows the code to be executed in parallel without any further changes being necessary.

params = {'ksp_type': 'preonly', 'pc_type': 'bjacobi', 'sub_pc_type': 'ilu'}
prob1 = LinearVariationalProblem(a, L1, dq)
solv1 = LinearVariationalSolver(prob1, solver_parameters=params)
prob2 = LinearVariationalProblem(a, L2, dq)
solv2 = LinearVariationalSolver(prob2, solver_parameters=params)
prob3 = LinearVariationalProblem(a, L3, dq)
solv3 = LinearVariationalSolver(prob3, solver_parameters=params)

We now run the time loop. This consists of three Runge-Kutta stages, and every 20 steps we write out the solution to file and print the current time to the terminal.

t = 0.0
step = 0
while t < T - 0.5*dt:
    solv1.solve()
    q1.assign(q + dq)

    solv2.solve()
    q2.assign(0.75*q + 0.25*(q1 + dq))

    solv3.solve()
    q.assign((1.0/3.0)*q + (2.0/3.0)*(q2 + dq))

    step += 1
    t += dt

    if step % 20 == 0:
        outfile.write(q)
        print("t=", t)

Finally, we display the normalised \(L^2\) error, by comparing to the initial condition.

L2_err = sqrt(assemble((q - q_init)*(q - q_init)*dx))
L2_init = sqrt(assemble(q_init*q_init*dx))
print(L2_err/L2_init)

This demo can be found as a script in DG_advection.py.

References

[LeV96]Randall J. LeVeque. High-Resolution Conservative Algorithms for Advection in Incompressible Flow. SIAM Journal on Numerical Analysis, 33(2):627–665, 1996. doi:10.1137/0733033.
[SO88]Chi-Wang Shu and Stanley Osher. Efficient Implementation of Essentially Non-oscillatory Shock-Capturing Schemes. Journal of Computational Physics, 77(2):439–471, 1988. doi:10.1016/0021-9991(88)90177-5.