In this paper, we study the superconvergence of the error for the discontinuous Galerkin (DG) finite element method for linear conservation laws when upwind fluxes are used. We prove that if we apply piecewise $k$-th degree polynomials, the error between the DG solution and the exact solution is ($k+2$)-th order superconvergent at the downwind-biased Radau points with suitable initial discretization. Moreover, we also prove the DG solution is ($k+2$)-th order superconvergent both for the cell averages and for the error to a particular projection of the exact solution. The superconvergence result in this paper leads to a new {\em a posteriori} error estimate. Our analysis is valid for arbitrary regular meshes and for $\mathcal{P}^k$ polynomials with arbitrary $k\geq1$, and for both periodic boundary conditions and for initial-boundary value problems. We perform numerical experiments to demonstrate that the superconvergence rate proved in this paper is optimal.

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We present a simple proof on the formation of flocking in the F. Cucker and S. Smale system [Jpn. J. Math. (3) 2, No. 1, 197–227 (2007; Zbl 1166.92323)] based on the explicit construction of a Lyapunov functional. Our results also provide a unified condition on the initial states in which the exponential convergence to the flocking state will occur. For large particle systems, we give a rigorous justification for the mean-field limit from the many particle Cucker-Smale system to the Vlasov equation with flocking dissipation as the number of particles goes to infinity.

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