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A Simple Explanation of Superconvergence for Discontinuous Galerkin Solutions to ut+ux=0

  • Philip Roe (a1)
Abstract
Abstract

The superconvergent property of the Discontinuous Galerkin (DG) method for linear hyperbolic systems of partial differential equations in one dimension is explained by relating the DG method to a particular continuous method, whose accuracy depends in part on a local analysis, and in part on information transferred from upwind elements.

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*Corresponding author. Email address: philroe@umich.edu (P. L. Roe)
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Communicated by Chi-Wang Shu
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References
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[1] Adjerid S., Devine K.D., Flaherty J.D., Krivodonova L., 2002, A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems, Computer methods in Applied Mechanics and Engineering, 191(11-12), pp. 10971112.
[2] Ainsworth M., 2004, Dispersive and dissipative behavior of high order discontinuous Galerkin finite-element methods, Journal of Computational Physics, 198, pp. 106130.
[3] Bouche D., Bonnaud G. and Ramos D., 2003. Comparison of numerical schemes for solving the advection equation. Applied mathematics letters, 16(2), pp.147154.
[4] Cao W., Zhang Z. and Zou Q., 2014. Superconvergence of discontinuous Galerkin methods for linear hyperbolic equations. SIAM Journal on Numerical Analysis, 52(5), pp.25552573.
[5] Hedstrom G.W., 1975. Models of difference schemes for ut +ux = 0 by partial differential equations. Mathematics of Computation, 29(132), pp.969977.
[6] Hu F.Q., Hussaini M.Y. and Rasetarinera P., 1999. An analysis of the discontinuous Galerkin method for wave propagation problems. Journal of Computational Physics, 151(2), pp.921946.
[7] Krivodonova L., Qin R., 2013, An analysis of the spectrum of the discontinuous Galerkin method. Applied Numerical Mathematics, 64, pp. 118.
[8] Meng X., Shu C-W., Wu B., 2015, Optimal error estimates for discontinuous Galerkin methods based on upwind-biased fluxes for linear hyperbolic equations.Math Comp, 85, pp. 12251261.
[9] Yang Y., and Shu C-W., 2012. Analysis of optimal superconvergence of discontinuous Galerkin method for linear hyperbolic equations. SIAM Journal on Numerical Analysis, 50(6), pp. 31103133.
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Communications in Computational Physics
  • ISSN: 1815-2406
  • EISSN: 1991-7120
  • URL: /core/journals/communications-in-computational-physics
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