Solving 1D inviscid Burger's equation with time dependent BC

Hello,

I am trying to solve 1D inviscid Burger's equation with time dependent BC (i.e. `u(x=0,t) = sin(2 pi t)`), while initially everywhere is zero i.e. `u(x,t=0) = 0`. However, the solution (`t=1`) is non-physical as shown below:

The codes is shown as follows:

```
using Trixi, OrdinaryDiffEq, Plots

function inlet_sine_bc(u_inner, orientation, direction, x, t, surface_flux::FluxLaxFriedrichs, equations::InviscidBurgersEquation1D)
return SVector(sin(2π * t))
end

equations = InviscidBurgersEquation1D()

solver = DGSEM(polydeg=3, surface_flux=flux_lax_friedrichs)
initial_condition(x, t, equations) = 0.0

boundary_conditions = (
x_neg = inlet_sine_bc,
x_pos = Trixi.BoundaryConditionDoNothing(),
)

semi = SemidiscretizationHyperbolic(
TreeMesh(0.0, 1.0, initial_refinement_level=5, n_cells_max = 10^4, periodicity = false),
equations, initial_condition, solver;
boundary_conditions = boundary_conditions
)

tspan = (0.0, 1.0)
ode = semidiscretize(semi, tspan)

controller = StepsizeCallback(cfl = 0.25)
save_times = 0:0.01:1.0
sol = solve(ode, SSPRK33(), dt=1e-3, saveat=save_times, callback=controller)

plot(sol)
```

Any comments are appreciated!

Run the provided Julia reproduction with Trixi, OrdinaryDiffEq, and Plots, then inspect the handling of `inlet_sine_bc`, `BoundaryConditionDoNothing()`, and `InviscidBurgersEquation1D` with `DGSEM`. Compare the result against the expected boundary-driven solution and determine whether the behavior is physical or indicates a boundary-condition issue; completion would require a documented explanation or a reproducible fix.

Written by the indexing model from the issue text.