Symmetry Boundary conditions in Dolfin DG

1

Hello everyone,

I am using dolfin-dg to study an euler-like problem:

\partial_t q_j + \nabla_i F_{ij}=0 \\

where:

q = (\rho, \rho u_1 , \rho u_2)^T

F = (\rho \vec{u}, \rho\vec{u} \otimes \vec{u} + \rho I_{2\times 2})^T

I was solving the problem in a rectangular mesh of 30 by 15 with an inflow centered in the lower wall and outflow everywhere else:

boundary_original
boundary_original1072Ă—755 21.9 KB

The inflow conditions are symmetric with respect to the x = 0 axis. I am imposing boundary conditions with:

bcs = [DGDirichletBC(ds(INLET), gD_inlet),
       DGNeumannBC(ds(OUTFLOW), g_out)]

Therefore my question is, is there any way I can solve the problem just in one half of the domain (say x>0) to improve spatial resolution without increasing computation time like they do in here? The domain should look like:

boundary_sym
boundary_sym1072Ă—755 35.5 KB

with the left wall beeing a symmetry plane. I think the boundary conditions there should be zero horizontal velocity and \frac{\partial \rho}{\partial y} = 0. But I am not sure of how to implement this.

Thanks.

2

There are a few things to consider:

  • Your problem is a first order PDE, so you’ll be over prescribing boundary data with a condition like \frac{\partial \rho}{\partial y} = 0 on some boundary. The link you’ve provided isn’t applicable for your case.
  • If the facets marked in red are outlet BCs, and your inlet condition is well behaved, I expect your solution to be symmetric about x = 0.
  • It may still possible to prescribe \vec{u} \cdot \vec{n} = 0 on the left boundary, depending on your inlet condition.
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