The von Neumann Stability Analysis of the Fixed‐Stress Schemes in Poroelastodynamics

ABSTRACT We investigate splitting schemes based on the fixed‐stress sequential approach for poroelastodynamic problems. To assess numerical stability, we perform the von Neumann stability analysis on several fixed‐stress schemes for poroelastodynamics, including staggered, stabilized, and iterative methods. Our analysis reveals that while the staggered fixed‐stress method is unconditionally stable for poroelastostatics, this unconditional stability does not extend to poroelastodynamics due to the presence of the second‐order time derivative in the geomechanics wave equation. Specifically, the staggered method becomes unstable when the Courant number falls below unity, indicating a lower bound on the time step size required for stability in poroelastodynamic simulations. The stabilized method, which incorporates an additional stabilization term, maintains numerical stability across the full range of Courant numbers. However, it suffers from limited convergence and reduced accuracy, particularly at low Courant numbers. In contrast, the iterative fixed‐stress method consistently converges to the monolithic solution, regardless of the Courant number, provided that full iteration is performed. Numerical tests validate these stability estimates and confirm agreement with the von Neumann stability analysis.