# A Biologically Motivated Finite Difference Approach for Simulating Singularly Perturbed Vertical Motion in Human Gait

## TL;DR

This paper introduces a novel finite difference numerical method, inspired by biology, for accurately simulating the vertical motion in human gait models involving singularly perturbed differential equations, addressing boundary layer challenges.

## Contribution

The study develops a domain decomposition and time-rescaling based finite difference scheme that effectively solves singularly perturbed gait equations with second-order accuracy.

## Key findings

- The method achieves second-order spatial accuracy.
- Numerical experiments confirm stability and efficiency.
- Framework applicable to complex biomechanical models.

## Abstract

In this study, we present a simulation-based numerical method for solving a class of singularly perturbed second-order differential equations that come from a simplified biologically motivated model of human gait. Important physical factors such as gravity, damping, and leg stiffness are included in the model, which also depicts the vertical motion of the center of mass of the body during walking or running. Most of the time, standard numerical methods are ineffective in resolving boundary layer behavior that occurs due to the small perturbation parameter in the governing equation. We use a domain decomposition technique to divide the problem domain into inner and outer regions to tackle this difficulty. The boundary layer resolves the steep gradients. We applied a time-rescaling transformation to the inner region. Each subdomain is discretized, and the resulting tridiagonal systems are efficiently solved using the Thomas algorithm within the mixed finite difference framework. A detailed convergence analysis demonstrates second-order accuracy in space. The numerical results validate the proposed scheme's accuracy, stability, and efficiency through experiments based on modified human gait models. The framework serves as a fundamental tool for biomechanical simulation. The modeling is a foundation for future research, incorporating nonlinearities, time delays, and real-world scenarios data on how people walk.