Summation-by-parts in time

Jan Nordström, Tomas Lundquist

Research output: Contribution to journalArticlepeer-review

64 Citations (Scopus)


We develop a new high order accurate time-integration technique for initial value problems. We focus on problems that originate from a space approximation using high order finite difference methods on summation-by-parts form with weak boundary conditions, and extend that technique to the time-domain. The new time-integration method is global, high order accurate, unconditionally stable and together with the approximation in space, it generates optimally sharp fully discrete energy estimates. In particular, it is shown how stable fully discrete high order accurate approximations of the Maxwells' equations, the elastic wave equations and the linearized Euler and Navier-Stokes equations can obtained. Even though we focus on finite difference approximations, we stress that the methodology is completely general and suitable for all semi-discrete energy-stable approximations. Numerical experiments show that the new technique is very accurate and has limited order reduction for stiff problems.

Original languageEnglish
Pages (from-to)487-499
Number of pages13
JournalJournal of Computational Physics
Publication statusPublished - 15 Oct 2013
Externally publishedYes


  • Boundary conditions
  • Convergence
  • Global methods
  • High order accuracy
  • Initial value boundary problems
  • Initial value problems
  • Stability
  • Stiff problems
  • Summation-by-parts operators
  • Time integration

ASJC Scopus subject areas

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • General Physics and Astronomy
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics


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