High-dimensional data acquisition in edge-computing IoT networks faces bottlenecks owing to constrained sampling rates, limited battery capacity, and numerical instabilities caused by signal stiffness. Continuous-domain ordinary differential equations (ODEs) enhance sparse recovery, but direct execution on digital microcontrollers remains computationally prohibitive. To fill this theoretical-to-hardware gap, we developed a lightweight discretization framework and evaluated six explicit and semi-implicit Euler algorithms derived from exponentially, finite-time, and fast fixed-time convergent ODEs, comparing with the iterative shrinkage-thresholding algorithm (ISTA) and fast ISTA. By incorporating semi-implicit discretizations, numerical oscillations were eliminated without the parameter overhead of fully implicit schemes. Simulation results on 512-dimensional sparse signals showed that semi-implicit fixed-time and finite-time algorithms achieved complete convergence (log10|vk| < −15) within 150 to 200 iterations, compared with more than 700 iterations required by standard ISTA. Explicit methods such as Finite-Exp prematurely plateaued near log10|vk| ≈ −4. Furthermore, Fix-Imp demonstrated superior step-size resilience under parameter variations (h = 0.05 to 0.3). By significantly reducing edge CPU cycles and maintaining rapid convergence, the developed framework advances energy-efficient, deterministic, and self-aware sensor technology for real-time industrial IoT deployments.