Numerical Simulation of Variational Dissipative Conservative Partial Differential Equation Systems

This deposit contains the paper “Numerical Simulation of Variational Dissipative–Conservative Partial Differential Equation Systems” by Kevin Ellis Anderson. The paper presents direct numerical simulations of a class of coupled variational partial differential equation systems generated by a single free-energy functional. The framework combines gradient-flow evolution for a dissipative deviation field with Hamiltonian (unitary) evolution for a conservative oscillatory field, allowing irreversible dissipation and conservative dynamics to coexist within a unified formulation. Using finite-difference schemes in one spatial dimension, the simulations demonstrate monotonic free-energy decay, global stability of solutions, irreversible energy transfer from oscillatory to dissipative degrees of freedom, and bounded oscillatory evolution. These numerical results confirm analytical well-posedness and Lyapunov properties established in earlier work and provide a computational foundation for further studies of localisation and event-like behaviour.