Interaction of Waves with Frictional Interfaces Using Summation-by-Parts Difference Operators: Weak Enforcement of Nonlinear Boundary Conditions

• Published in: Journal of scientific computing Vol. 50; no. 2; pp. 341 - 367
• Main Authors: Kozdon, Jeremy E., Dunham, Eric M., Nordström, Jan
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.02.2012; Springer Nature B.V
• Subjects: Algorithms; Applied mathematics; Boundary conditions; Computational Mathematics and Numerical Analysis; Discretization; Earthquakes; Elastodynamics; Finite differences; Finite element method; High order finite difference · Nonlinear boundary conditions · Simultaneous; Interfaces; MATEMATIK; Mathematical and Computational Engineering; Mathematical and Computational Physics; MATHEMATICS; Mathematics and Statistics; Numerical methods; Operators (mathematics); Seismic waves; Sliding; Strain energy; Theoretical; Tillämpad matematik; Time integration; Nonlinear boundary conditions; Wave propagation; Friction; Summation-by-parts; Simultaneous approximation term method; Elastodynamics; High order finite difference
• ISSN: 0885-7474; 1573-7691; 1573-7691
• DOI: 10.1007/s10915-011-9485-3

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.