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Stationary discrete shock profiles for scalar conservation laws with a discontinuous Galerkin method

We present an analysis of stationary discrete shock profiles for a discontinuous Galerkin method approximating scalar nonlinear hyperbolic conservation laws with a convex flux. Using the Godunov method for the numerical flux, we characterize the steady state solutions for arbitrary approximation orders and show that they are oscillatory only in one mesh cell and are parametrized by the shock strength and its relative position in the cell. In the particular case of the inviscid Burgers equation, we derive analytical solutions of the numerical scheme and predict their oscillations up to fourth-order of accuracy. Moreover, a linear stability analysis shows that these profiles may become unstable at points where the Godunov flux is not differentiable. Theoretical and numerical investigations show that these results can be extended to other numerical fluxes. In particular, shock profiles are found to vanish exponentially fast from the shock position for some class of monotone numerical fluxes and the oscillatory and unstable characters of their solutions present strong similarities with that of the Godunov method.