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

BACKGROUND AND

OBJECTIVE

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.

METHODS

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.

RESULTS

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 Δxcrit=2.2×10-6 m and Δtcrit=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.

CONCLUSION

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.