Journal of Scientific Computing

, Volume 79, Issue 3, pp 1572–1607 | Cite as

Error Boundedness of Discontinuous Galerkin Methods with Variable Coefficients

  • Philipp ÖffnerEmail author
  • Hendrik Ranocha


For practical applications, the long time behaviour of the error of numerical solutions to time-dependent partial differential equations is very important. Here, we investigate this topic in the context of hyperbolic conservation laws and flux reconstruction schemes, focusing on the schemes in the discontinuous Galerkin spectral element framework. For linear problems with constant coefficients, it is well-known in the literature that the choice of the numerical flux (e.g. central or upwind) and the selection of the polynomial basis (e.g. Gauß–Legendre or Gauß–Lobatto–Legendre) affects both the growth rate and the asymptotic value of the error. Here, we extend these investigations of the long time error to variable coefficients using both Gauß–Lobatto–Legendre and Gauß–Legendre nodes as well as several numerical fluxes. We derive conditions guaranteeing that the errors are still bounded in time. Furthermore, we analyse the error behaviour under these conditions and demonstrate in several numerical tests similarities to the case of constant coefficients. However, if these conditions are violated, the error shows a completely different behaviour. Indeed, by applying central numerical fluxes, the error increases without upper bound while upwind numerical fluxes can still result in uniformly bounded numerical errors. An explanation for this phenomenon is given, confirming our analytical investigations.


Flux reconstruction Discontinuous Galerkin Summation-by-parts Error analysis Error bound Error growth 

Mathematics Subject Classification

65N15 65N12 65N35 65N30 65N06 



Philipp Öffner was supported by SNF Project (Number 175784) “Solving advection dominated problems with high order schemes with polygonal meshes: application to compressible and incompressible flow problems” and Hendrik Ranocha was supported by the German Research Foundation (DFG, Deutsche Forschungsgemeinschaft) under Grant SO 363/14-1.


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Universität ZürichZürichSwitzerland
  2. 2.TU BraunschweigBraunschweigGermany

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