# Space-time Galerkin of Burgers changes the convection speed

tldr: Can space-time Galerkin schemes applied to convection-diffusion problems lead to effects on the convection velocity?

For time $t\in (0,1)$ and the spatial variable $\xi \in (0,1)$, I am considering the viscous Burger's equation

$$\dot z + \frac 12\partial_\xi (z^2) - \nu \partial_{\xi\xi}z = 0 \tag{*}$$

with a viscosity parameter $\nu$. I have zero Dirichlet conditions in space and as the initial condition $z(0)$, I use a step function, which is one on one half of the domain and zero on the other half.

With finite elements in space and Runge-Kutta in time, I get to the expected numerical results.

Now, if I discretize the equation with a space-time Galerkin scheme, I get a picture, where, apart from artefacts due to low-dimensional bases, the convection speed is obviously too fast. (The front reaches the border too early)

In fact, if I replace the factor $\frac 12$ by the factor $0.375$ in the equation $(*)$, then the convection looks right.

Since this factor $0.375$ works for all kind of experiments, namely for various viscosity parameters $\nu$ and sizes of the space-time bases, I am almost sure that this is not a bug in my code. Thus, my question is: Is there a known issue concerning space-time discretizations of transient convection-diffusion problems that cause this corrections in the convection speed?

Note that I use continuous Galerkin in time as well, so that there is no interpretation of the time-space scheme, as a space discretization plus time stepping.

Edit (of the first edit): Here is the result of the space-time Galerkin simulation for a finer resolution, a smooth initial condition that is consistent with the Dirichlet conditions, and a shorter time frame to avoid the "clash" of the wave with the zero Dirichlet conditions:

As before, in the low-order space-time Galerkin approach (right) the wave speed is too high if compared to the FEM/Runge-Kutta solution (left).

EDIT (and confession) I must admit that it was an error in the implementation. After @BillBarth's answer and after that Joost from nutils provided me a benchmark simulation of the same problem with high-fidelity space-time finite elements I was convinced to revise my code bottom-up. Anyways, this is how it could look like with a low-dimensional (12, 12) degrees of freedom in (space, time) space time Galerkin scheme.

• You're still pretty unresolved in $x$. I can't guarantee that this is your problem, but I don't think you get to ask until you get rid of the wiggles. To quote Gresho "they're telling you something". – Bill Barth Jun 24 '16 at 13:45