Under a polynomial reconstruction hypothesis, entropy-stable DG methods converge at O(h^k) in a norm equivalent to L^2.
Chandrashekar,Kinetic energy preserving and entropy stable finite volume schemes for compressible Euler and Navier–Stokes equations, Commun
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The error estimate of entropy-stable discontinuous Galerkin methods for hyperbolic conservation laws
Under a polynomial reconstruction hypothesis, entropy-stable DG methods converge at O(h^k) in a norm equivalent to L^2.