Dedicated to Professor Stanley Osher on the occasion of his 70th birthday.
In this paper, we establish negative-order norm estimates for the accuracy of discontinuous Galerkin (DG) approximations to scalar nonlinear hyperbolic equations with smooth solutions. For these special solutions, we are able to extract this “hidden accuracy” through the use of a convolution kernel that is composed of a linear combination of B-splines. Previous investigations into extracting the superconvergence of DG methods using a convolution kernel have focused on linear hyperbolic equations. However, we now demonstrate that it is possible to extend the Smoothness-Increasing Accuracy-Conserving filter for scalar nonlinear hyperbolic equations. Furthermore, we provide theoretical error estimates for the DG solutions that show improvement to \((2k+m)\)-th order in the negative-order norm, where \(m\) depends upon the chosen flux.
Adjerid, S., Baccouch, M.: Asymptotically exact a posteriori error estimates for a one-dimensional linear hyperbolic problem. Appl. Numer. Math.
60, 903–914 (2012)
Adjerid, S., Baccouch, M.: The discontinous Galerkin error estimation for two-dimensional hyperbolic problems II: A posterior error estimation. J. Sci. Comput.
38, 15–49 (2009)
Adjerid, S., Baccouch, M.: The discontinous Galerkin error estimation for two-dimensional hyperbolic problems I: superconvergence error analysis. J. Sci. Comput.
33, 75–113 (2007)
Bramble, J.H., Schatz, A.H.: Higher order local accuracy by averaging in the finite element method. Math. Comput.
31, 94–111 (1977)
Bramble, J.H., Schatz, A.H., Thomée, V., Wahlbin, L.B.: Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equations. SIAM J. Numer. Anal.
14, 218–241 (1977)
Brenner, S.C.: The Mathematical Theory of Finite Element Methods. Springer, New York (2002)
Ciarlet, P.: The finite element method for elliptic problem. North Holland (1975)
Cockburn, B., Luskin, M., Shu, C.-W., Süli, E.: Enhanced accuracy by post-processing for finite element methods for hyperbolic equations. Math. Comput.
72, 577–606 (2003)
Cockburn, B., Shu, C.-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Math. Comput.
52, 411–435 (1989)
Hörmander, L.: Lectures on Nonlinear Hyperbolic Differential Equations. Springer, Berlin (1997)
Ji, L., Xu, Y., Ryan, J.K.: Accuracy-enhancement of discontinuous Galerkin solutions for convection-diffusion equations in multiple-dimensions. Math. Comput.
81, 1929–1950 (2012)
Majda, A., Osher, S.: Propagation of error into regions of smoothness for accurate difference approximate solutions to hyperbolic equations. Comm. Pure Appl. Math.
30, 671–705 (1977)
Marchuk, G.I.: Construction of adjoint operators in non-linear problems of mathematical physics. Sbornik Math.
189, 1505–1516 (1998)
Mirzaee, H., Ji, L., Ryan, J.K., Kirby, R.M.: Smoothness-increasing accuracy-conserving (SIAC) post-processing for discontinuous Galerkin solutions over structured triangular meshes. SIAM J. Numer. Anal.
49, 1899–1920 (2011)
Mock, M.S., Lax, P.D.: The computation of discontinuous solutions of linear hyperbolic equations. Comm. Pure Appl. Math.
31, 423–430 (1978)
van Slingerland, P., Ryan, J.K., Vuik, C.W.: Position-dependent smoothness-increasing accuracy-conserving (SIAC) filtering for accuracy for improving discontinuous Galerkin solutions. SIAM J. Sci. Comput.
33, 802–825 (2011)
Steffan, M., Curtis, S., Kirby, R.M., Ryan, J.K.: Investigation of smoothness enhancing accuracy-conserving filters for improving streamline integration through discontinuous fields. IEEE-TVCG.
14, 680–692 (2008)
Thomée, V.: High order local approximations to derivatives in the finite element method. Math. Comput.
31, 652–660 (1977)
Xu, Y., Shu, C.-W.: Error estimates of the semi-discrete local discontinuous Galerkin method for nonlinear convection-diffusion and KdV equations. Comput. Methods Appl Mech. Eng.
196, 3805–3822 (2007)
Yan, J., Shu, C.-W.: A local discontinuous Galerkin method for KdV type equations. SIAM J. Numer. Anal.
40, 769–791 (2002)
Zhang, Q., Shu, C.-W.: Error estimates to smooth solutions of Runge-Kutta discontinuous Galerkin methods for scalar conservation laws. SIAM J. Numer. Anal.
42, 641–666 (2004)
Zhang, Q., Shu, C.-W.: Error estimates to smooth solutions of Runge-Kutta discontinuous Galerkin methods for symmetrizable systems of conservation laws. SIAM J. Numer. Anal.
44, 1703–1720 (2006)
Zhang, Q., Shu, C.-W.: Stability analysis and a priori error estimates of the third order explicite Runge-Kutta discontinuous Galerkin methods for scalar conservation laws. SIAM J. Numer. Anal.
48, 1038–1063 (2010)
About this article
Negative-Order Norm Estimates for Nonlinear Hyperbolic Conservation Laws