Numerical Stability of Soil‐Water Coupled Analysis With Implicit Time Integration Using Spectral Radius of Recursive Equation Comprising Discretized Governing Equation of <i> <b>u</b> ‐p </i> Formulation

ABSTRACT The numerical stability of the u ‐ p formulation‐based dynamic soil‐water coupled analysis was evaluated using the spectral radius of a simultaneous recursive equation derived from the spatiotemporally discretized governing equations and time‐integration formulas. The findings reveal that the unstable range of the u ‐ p formulation forms a band‐shaped region in the permeability coefficient k ‐time increment Δ t space. This band bends at the point where the theoretical solution of the u ‐ p formulation transitions. By analyzing the transience of the eigenvector along the band, we identified significant eigenvector rotation around the bend, suggesting that the differing appearances of unstable regions between low‐ and high‐permeability ranges are caused by variations in the divergence of eigenmodes. Moreover, the criterion proposed by Noda et al. (2008) for evaluating the feasibility of u ‐ p formulation‐based analysis, which relies on the ratio k /Δ t , is generally valid for low‐permeability ranges. However, this criterion proves inconsistent for high‐permeability ranges, where feasibility depends solely on Δ t .