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Divided difference estimates and accuracy enhancement of discontinuous Galerkin methods for nonlinear symmetric systems of hyperbolic conservation laws. (English) Zbl 1412.65153

The paper is concerned with the accuracy enhancement of the discontinuous Galerkin method for one-dimensional nonlinear systems of conservation laws. The authors study properties of the discontinuous Galerkin discretization operator and properties of divided differences. For discretization they use the finite element spaces of piecewise polynomial functions of degree \(k\) on uniform meshes and the upwind fluxes. Under the assumption that the flux Jacobian matrix is positive definite and \(1 \leq \alpha \leq k-1\), the authors prove that the \(L^2\)-norm of the \(\alpha\)th-order divided difference of the error is of order \(k+3/2-\alpha/2\). Using the duality argument, they derive that the norm of order \(-k+1\) of the \(\alpha\)th-order divided difference of the error is of order \(2k+3/2-\alpha/2\). This implies that the convergence of the post-processed solution with respect to the \(L^2\)-norm is of order \(3k/2+1\). Numerical experiments are presented to confirm the theoretical results.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
35Q31 Euler equations
86A10 Meteorology and atmospheric physics
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