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.

FEM in space + Runge-Kutta in time

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)

low-rank space-time Galerkin with 12*12 basis functions

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

low-rank space-time Galerkin with corrected convection speed with 12*12 basis functions

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:

full solution and space-time Galerkin with 20*18 basis functions

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.

enter image description here


1 Answer 1


In your third picture, there are still a lot of spurious oscillations. You clearly know your mesh is under-resolved by at least a factor of 2. I'm guessing that the factor of 0.5 in the second term of (*) is externally imposed and that you shouldn't be arbitrarily changing it. Each space-time Galerkin method is equivalent to a finite difference scheme in space and time, depending on your choice of quadrature. I don't know what quadrature you've done here, and I don't have the equivalences memorized, but if this doesn't work out to something that's symplectic, you're very likely to go too fast or too slow.

I'd recommend refining the mesh until the oscillations stop first and see if you pick up the required wave speed. If not, you should work backwards and from one timestep (i.e. one time test function) of your space-time Galerkin method with whatever quadrature method you are using and see what FDM you have. It's probably not sympletic and is likely messing up everything.

  • $\begingroup$ Thank you for having a look into it. I have added a picture for a finer space-time grid (now 25*25 instead of 12*12 basis functions) to make the point that the effect is not due to underresolution. On the conceptual side; I really don't think that I need a scheme that resembles a symplectic integrator, since such integrators will mainly pay off in terms of long time energy preservation. They may have a smaller consistency error locally but that can be compensated by finer discretizations. Also, the initial simulation (first picture) did not resort to a symplectic time integrator. $\endgroup$
    – Jan
    Jun 24, 2016 at 6:25
  • $\begingroup$ 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". $\endgroup$
    – Bill Barth
    Jun 24, 2016 at 13:45
  • $\begingroup$ That's right. I have done some more simulations in a more moderate setup (smooth initial condition, shorter time frame) and I see the same picture of the incorrect wave speed... $\endgroup$
    – Jan
    Jun 27, 2016 at 8:34

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