Unified stability conditions for explicit finite-difference Biot-type poroelastodynamics: Non-dimensional design maps for time step selection in brain fluid transport.

Abstract

<h4>Background and objective</h4>Biot-type poroelastodynamics describes coupled solid deformation and pore-fluid transport, and is increasingly germane to biomedical simulation of tissue mechanics and interstitial fluid motion. Yet explicit finite-difference schemes, despite their economy in memory and execution time, remain hampered by the absence of a unified stability condition for systems in which propagative and dissipative processes coexist. This study derives a framework for explicit time-step selection in the u→-p form of Biot's equations.<h4>Methods</h4>A one-dimensional reduction of the linear Biot poroelastodynamic model was discretised with centred spatial differences, central differencing for the second-order displacement time derivative, and forward differencing for first-order pore-pressure evolution. Von Neumann analysis was then applied to the coupled discrete system, yielding a cubic characteristic polynomial in the amplification factor. By separating diffusion-dominated and wave-dominated asymptotic limits and invoking Schur stability theory, explicit admissibility bounds were obtained in terms of material coefficients, mesh size, time step, and wavenumber. A reformulation exposed the governing scales and enabled the construction of a Δx-Δt stability map. The derived conditions were then examined through root-locus analysis and healthy-brain simulations.<h4>Results</h4>The analysis yields two complementary stability restrictions: a diffusion-like condition, Δt/Δx²≤1/(2D), and a wave-like condition, Δt/Δx≤1/C, showing that the explicit scheme cannot be characterised adequately by a single universal condition. Their intersection defines a critical mesh size and critical time step, with representative brain parameters giving Δx_crit=2.2×10^-6 m and Δt_crit=6.879×10^-8 s. This crossover identifies the scale at which stability control passes from wave transmission across one grid interval to diffusive pore-pressure relaxation over the same interval. Spectral analysis further reveals distinct failure modes: a real-root-dominated, non-oscillatory breakdown in the diffusion-dominated regime and a conjugate-pair-driven oscillatory instability in the wave-dominated regime. Numerical experiments corroborate the theoretical boundaries and demonstrate bounded physiological responses only when both mesh and time step lie within the admissible region.<h4>Conclusion</h4>The resulting framework furnishes parameter-aware design rules for explicit poroelastodynamic computation and offers a basis for brain fluid-transport simulation, with broader relevance to other Biot-type porous media systems cast in the same coupled displacement-pressure formulation.