In your second 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.

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.