 # BCs needed for unique solution to compressible Euler equation

Hi, I was able to simulate the compressible Euler equations for a single phase dilute particle cloud subject to body forces as described in:

I used cylindrical coordinates. The form I used is the following:

r = Expression('x', degree=deg)
F = (c*(u-u_1) + u*(c-c_1))*r*k_t*v_u*dx + (c*u*u)*v_u*dx + (c*u*u).dx(0)*r*v_u*dx - \
F_drag(u,c)*r*k_mp*v_u*dx - c*F_applied*r*k_mp*v_u*dx + \
(c - c_1)*r*k_t*v_c*dx + (c*u)*v_c*dx + (c*u).dx(0)*r*v_c*dx


The initial conditions are a Gauss function for the initial particle concentration c, an initial velocity u of the form u_0 = k*x and a force field pushing particles towards the center and decreasing to zero at the center.

The simulation gives reasonable results, however it does not need any boundary conditions. I am not sure why this is the case.

My reasoning of why this works is that there is mass conservation for c that keeps the area under the curve constant (therefore no Dirichlet BCs are needed). The u field does not need any Dirichlet BCs because the speed is zero at x=0 because F = 0 there. I added u =0 at x=0 as a BC and get the same result.

Is the above reasoning correct? What about natural BCs? I understand why there might be natural Von Neumann BCs:
https://fenicsproject.org/docs/dolfin/1.6.0/python/demo/documented/neumann-poisson/python/documentation.html
And based on the observed results this might apply for c but what about u?

Alex

I’m not really sure exactly what you’re doing as I’m unable to get access to the paper, however:

It makes no sense for an advection (no diffusion) problem to have Neumann BCs. By the method of characteristics you can see that you need to treat a numerical solution approximation carefully at the boundaries by considering the fluxes. These flux data are enforced in a different mathematical setting to the standard FEM discretisation of a second order PDE.

As for why you “dont need BCs” for your time dependent problem. You can think of your domain as being \Omega \times (0, T], where \Omega is your spatial domain, and (0, T] is the temporal domain. The boundary condition is the initial condition at t=0. This defines the initial fluxes into your spacetime domain.

Hi Nate, thank you for the prompt reply.

For the second point about not needing an BCs, do you have any references/books I could go over explaining this in more detail?

Any introduction to analysis or numerical methods for PDEs. Specifically hyperbolic PDEs.

LeVeque’s book is a classic.