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Superconvergence of the space-time discontinuous Galerkin method for linear nonhomogeneous hyperbolic equations. (English) Zbl 1480.65259

Summary: In this study, we discuss the superconvergence of the space-time discontinuous Galerkin method for the first-order linear nonhomogeneous hyperbolic equation. By using the local differential projection method to construct comparison function, we prove that the numerical solution is \((2n+1)\)-th order superconvergent at the downwind-biased Radau points in the discrete \(L^2\)-norm. As a by-product, we obtain a point-wise superconvergence with order \(2n+\frac{1}{2}\) in vertices. We also find that, in order to obtain these superconvergence results, the source integral term has to be approximated by \((n+1)\)-point Radau-quadrature rule. Numerical results are presented to verify our theoretical findings.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds 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
65D32 Numerical quadrature and cubature formulas
35L02 First-order hyperbolic equations
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