Interaction of waves with frictional interfaces using summation-by-parts difference operators: Weak enforcement of nonlinear boundary conditions

Jeremy E. Kozdon, Eric M. Dunham, Jan Nordström

Research output: Contribution to journalArticlepeer-review

49 Citations (Scopus)


We present a high-order difference method for problems in elastodynamics involving the interaction of waves with highly nonlinear frictional interfaces. We restrict our attention to two-dimensional antiplane problems involving deformation in only one direction. Jump conditions that relate tractions on the interface, or fault, to the relative sliding velocity across it are of a form closely related to those used in earthquake rupture models and other frictional sliding problems. By using summation-by-parts (SBP) finite difference operators and weak enforcement of boundary and interface conditions, a strictly stable method is developed. Furthermore, it is shown that unless the nonlinear interface conditions are formulated in terms of characteristic variables, as opposed to the physical variables in terms of which they are more naturally stated, the semi-discretized system of equations can become extremely stiff, preventing efficient solution using explicit time integrators. The use of SBP operators also provides a rigorously defined energy balance for the discretized problem that, as the mesh is refined, approaches the exact energy balance in the continuous problem. This enables one to investigate earthquake energetics, for example the efficiency with which elastic strain energy released during rupture is converted to radiated energy carried by seismic waves, rather than dissipated by frictional sliding of the fault. These theoretical results are confirmed by several numerical tests in both one and two dimensions demonstrating the computational efficiency, the high-order convergence rate of the method, the benefits of using strictly stable numerical methods for long time integration, and the accuracy of the energy balance.

Original languageEnglish
Pages (from-to)341-367
Number of pages27
JournalJournal of Scientific Computing
Issue number2
Publication statusPublished - Feb 2012
Externally publishedYes


  • Elastodynamics
  • Friction
  • High order finite difference
  • Nonlinear boundary conditions
  • Simultaneous approximation term method
  • Summation-by-parts
  • Wave propagation

ASJC Scopus subject areas

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


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