Comparison of fault representation methods in finite difference simulations of dynamic rupture

Assessing accuracy of numerical methods for spontaneous rupture simulation is challenging because we lack analytical solutions for reference. Previous comparison of a boundary integral method (BI) and finite-difference method (called DFM) that explicitly incorporates the fault discontinuity at velocity nodes (traction-at-split-node scheme) shows that both converge to a common, grid-independent solution and exhibit nearly identical power-law convergence rates with respect to grid spacing Delta x. We use this solution as a reference for assessing two other proposed finite-difference methods, the thick fault (TF) and stress glut (SG) methods, both of which approximate the fault-jump conditions through inelastic increments to the stress components (inelastic-zone schemes). The TF solution fails to match the qualitative rupture behavior of the reference solution and has quantitative misfits in root-mean-square rupture time of similar to 30% for the smallest computationally feasible Delta x (with similar to 9 grid-point resolution of cohesive zone, denoted (N) over bar (c) = 9). For sufficiently small values of Delta x, the SG method reproduces the qualitative features of the reference solution, but rupture velocity remains systematically low for SG relative to the reference solution, and SG lacks the well-defined power-law convergence seen for BI and DFM. The rupture-time error for SG, with (N) over bar (c) similar to 9, remains well above uncertainty in the reference solution, and the split-node method attains comparable accuracy with (N) over bar (c) 1/4 as large (and computation timescales as ((N) over bar (c))(4)). Thus, accuracy is highly sensitive to the formulation of the fault-jump conditions: The split-node method attains power-law convergence. The SG inelastic-zone method achieves solutions that are qualitatively meaningful and quantitatively reliable to within a few percent, but convergence is uncertain, and SG is computationally inefficient relative to the split-node approach. The TF inelastic-zone method does not achieve qualitatively meaningful solutions to the 3D test problem and is sufficiently computationally inefficient that it is not feasible to explore convergence quantitatively.