ENERGY-BASED MODELING AND STRUCTURE-PRESERVING DISCRETIZATION OF PHYSICAL SYSTEMS

Abstract

This paper develops a unified energy-based framework for modeling and discretizing physical systems, extending port-Hamiltonian theory to handle constraints and differential-algebraic equations with greater flexibility. We introduce a generalized approach that incorporates algebraic variables while preserving the core dissipation inequality, alongside two structure-preserving discretization methods: the midpoint rule for quadratic Hamiltonians and discrete gradients for nonlinear systems—both maintaining discrete energy balance and compatible with power-conserving networks. Our key contributions include: (i) structure-preserving model reduction with provable error bounds and equilibrium preservation; (ii) robustness analysis under parameter perturbations; (iii) adaptive time-stepping with optimal convergence; (iv) global existence results for constrained systems; and (v) second-order accuracy with fourth-order energy super-convergence. These advances are demonstrated across mechanical, electrical, thermal-fluid, and multi-physics applications, delivering reliable numerical tools that faithfully capture essential physical energy properties.