This paper aims to study, from a numerical point of view, a system of partial differential equations defined by a parabolic, an elliptic and a hyperbolic equation that can be used to describe cell migration in elastic tissues that integrates chemotaxis, interstitial fluid pressure and tissue displacement. The system couples a Keller-Segel type system for cell density and chemical concentration with an elliptic equation for fluid pressure and a wave-type equation for displacement. We propose a semidiscrete nonuniform finite difference method to approximate the continuous system.
Furthermore, we derive second-order spatial error estimates for all system variables using a suitable discrete norm. Finally, numerical simulations validate the theoretical results and demonstrate the interplay between chemotaxis and tissue mechanics.