Space-time <PERSON> of <PERSON> changes the convection speed
tldr: Can space-time <PERSON> 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 <PERSON>'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 <PERSON> 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. 346
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 <PERSON> 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 <PERSON> 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 <PERSON>'s answer and after that <PERSON> 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 <PERSON> scheme. 346