CALCOLO

, Volume 37, Issue 4, pp 207-232

First online:

An hp a priori error analysis of¶the DG time-stepping method for initial value problems

  • D. SchötzauAffiliated withSchool of Mathematics, University of Minnesota, Vincent Hall, 206 Church Street SE, Minneapolis, MN 55455, USA¶e-mail: schoetza@math.umn.edu
  • , C. SchwabAffiliated withSeminar für Angewandte Mathematik, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland¶e-mail: schwab@sam.math.ethz.ch

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Abstract:

The Discontinuous Galerkin (DG) time-stepping method for the numerical solution of initial value ODEs is analyzed in the context of the hp-version of the Galerkin method. New a priori error bounds explicit in the time steps and in the approximation orders are derived and it is proved that the DG method gives spectral and exponential accuracy for problems with smooth and analytic time dependence, respectively. It is further shown that temporal singularities can be resolved at exponential rates of convergence if geometrically refined time steps are employed.