Accuracy of Stable, High-order Finite Difference Methods for Hyperbolic Systems with Non-smooth Wave Speeds

Brittany A. Erickson, Ossian O’Reilly, Jan Nordström

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)


We derive analytic solutions to the scalar and vector advection equation with variable coefficients in one spatial dimension using Laplace transform methods. These solutions are used to investigate how accuracy and stability are influenced by the presence of discontinuous wave speeds when applying high-order-accurate, skew-symmetric finite difference methods designed for smooth wave speeds. The methods satisfy a summation-by-parts rule with weak enforcement of boundary conditions and formal order of accuracy equal to 2, 3, 4 and 5. We study accuracy, stability and convergence rates for linear wave speeds that are (a) constant, (b) non-constant but smooth, (c) continuous with a discontinuous derivative, and (d) constant with a jump discontinuity. Cases (a) and (b) correspond to smooth wave speeds and yield stable schemes and theoretical convergence rates. Non-smooth wave speeds [cases (c) and (d)], however, reveal reductions in theoretical convergence rates and in the latter case, the presence of an instability.

Original languageEnglish
Pages (from-to)2356-2387
Number of pages32
JournalJournal of Scientific Computing
Issue number3
Publication statusPublished - 1 Dec 2019
Externally publishedYes


  • High-order accuracy
  • Non-smooth wave speeds
  • Skew-symmetric splitting

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science
  • Numerical Analysis
  • General Engineering
  • Computational Theory and Mathematics
  • Computational Mathematics
  • Applied Mathematics


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