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We analyze discontinuous Galerkin methods using upwind-biased numerical fluxes for time-dependent linear conservation laws. In one dimension, optimal a priori error estimates of order k+1 are obtained for the semidiscrete scheme when piecewise polynomials of degree at most k (k ≥ 0) are used. Our analysis is valid for arbitrary nonuniform regular meshes and for both periodic boundary conditions and for initial-boundary value problems. We extend the analysis to the multidimensional case on Cartesian meshes when piecewise tensor product polynomials are used, and to the fully discrete scheme with explicit Runge-Kutta time discretization. Numerical experiments are shown to demonstrate the theoretical results.

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In this paper, the analysis of the superconvergence property of the discontinuous Galerkin (DG) method applied to one-dimensional time-dependent nonlinear scalar conservation laws is carried out. We prove that the error between the DG solution and a particular projection of the exact solution achieves (k + 3 2)-th order superconvergence when upwind fluxes are used. The results hold true for arbitrary nonuniform regular meshes and for piecewise polynomials of degree k (k ≥ 1), under the condition that |f ′ (u)| possesses a uniform positive lower bound. Numerical experiments are provided to show that the superconvergence property actually holds true for nonlinear conservation laws with general flux functions, indicating that the restriction on f (u) is artificial.

Cross-linked chitosan microspheres (40-100 microm) with smooth surface were prepared by the methods of emulsification and ethanol coagulant. FTIR results showed that the cross-linking reaction occurred on the amino groups of chitosan molecules. The swelling characteristic of chitosan microspheres was influenced by the environment pH, being generally greater at low rather than higher pH values. The coagulation properties of chitosan microspheres were evaluated by dynamic blood clotting, platelet adhesion and activation, erythrocyte adhesion, hemolysis, and protein absorption assays. Chitosan microspheres can shorten the clotting time and induce the adhesion and activation of platelets. But the shortening of clotting time by chitosan microspheres may be related to not only platelet aggregation, but also erythrocyte aggregation. Take together, chitosan microspheres may be potential use as thrombospheres.