A Thermodynamically Consistent Master-Equation Framework for Fluid Dynamics: Unified Discretization, Structure-Preserving Algorithms, and Convergence to Navier–Stokes Solutions

We propose a thermodynamically consistent framework for computational fluid dynamics based on a master equation formulation that precedes the continuum description. Instead of discretizing the Navier–Stokes equations, the fluid is modeled as a network of interacting elements whose dynamics are governed by antisymmetric conservative interactions and entropy-generating dissipative interactions. This construction embeds conservation laws and the second law of thermodynamics directly at the discrete level. A unified graph-based discretization is introduced, in which finite volume and meshless methods arise as special cases of a common interaction structure. Reversible fluxes are constructed to be entropy-neutral, while irreversible fluxes are derived from an entropy gradient flow, yielding a systematic decomposition of transport and dissipation. An implicit–explicit (IMEX) time integration scheme is then designed to preserve conservation and ensure entropy monotonicity. We establish convergence of the resulting scheme to entropy weak solutions of the Navier–Stokes equations through uniform a priori estimates and compactness arguments. Numerical experiments on compressible flow demonstrate that the proposed method maintains stability comparable to classical schemes while significantly reducing numerical dissipation, particularly in the resolution of contact discontinuities. These results suggest a shift in perspective: fluid dynamics can be understood not primarily as a system of partial differential equations, but as a thermodynamically constrained interaction system from which continuum equations emerge as effective limits. This viewpoint provides a unified foundation for the design of stable, accurate, and physically consistent numerical methods.