Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-23T15:53:03.325Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Hyperbolic equations and systems Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  08 March 2017

Philip Roe
Philip Roe*
Affiliation:
Department of Aerospace Engineering, University of Michigan, Ann Arbor 48109, USA
*
*Corresponding author. Email address:[email protected] (P. L. Roe)
Get access

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.

Keywords

Linear advectiondiscontinuous Galerkinsuperconvergence

MSC classification

Secondary: 35L03: Initial value problems for first-order hyperbolic equations 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 4 , April 2017 , pp. 905 - 912
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Communicated by Chi-Wang Shu

References

[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. 1097–1112.CrossRefGoogle Scholar
[2] Ainsworth, M., 2004, Dispersive and dissipative behavior of high order discontinuous Galerkin finite-element methods, Journal of Computational Physics, 198, pp. 106–130.CrossRefGoogle Scholar
[3] Bouche, D., Bonnaud, G. and Ramos, D., 2003. Comparison of numerical schemes for solving the advection equation. Applied mathematics letters, 16(2), pp.147–154.CrossRefGoogle Scholar
[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.2555–2573.CrossRefGoogle Scholar
[5] Hedstrom, G.W., 1975. Models of difference schemes for ut +ux = 0 by partial differential equations. Mathematics of Computation, 29(132), pp.969–977.Google Scholar
[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.921–946.CrossRefGoogle Scholar
[7] Krivodonova, L., Qin, R., 2013, An analysis of the spectrum of the discontinuous Galerkin method. Applied Numerical Mathematics, 64, pp. 1–18.CrossRefGoogle Scholar
[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. 1225–1261.CrossRefGoogle Scholar
[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. 3110–3133.CrossRefGoogle Scholar