Drift-Free Conservative Dynamics from Quantized Interaction Rules

TL;DR

This paper introduces an exact discrete operator formulation for conservative dynamics using quantized state spaces and antisymmetric integer-transfer operators, eliminating round-off errors and preserving invariants precisely.

Contribution

It presents a novel operator-level approach to discretize conservation laws exactly, avoiding flux-based approximations and round-off drift, applicable to scalar and multidimensional systems.

Findings

Exact conservation achieved at the arithmetic level.

High-frequency transport preserved near Nyquist limit.

Localized discontinuities maintained in Burgers dynamics.

Abstract

Conservation laws are conventionally discretized through floating-point flux evaluation, with invariants obtained by cancellation of approximate interface contributions and admissible weak solutions selected by reconstruction and Riemann solvers. Here we introduce an operator-level formulation in which conservative dynamics is realized as an exact discrete interaction rule on a quantized state space. The update is defined by an antisymmetric integer-transfer operator, which enforces conservation exactly at the arithmetic level and eliminates round-off drift from the primitive evolution \cite{highamAccuracyStabilityNumerical2002}. For scalar laws, monotone order-preserving transfers select admissible shock structures within the primitive update, rather than through flux reconstruction. Numerical experiments show that the interaction rule preserves high-frequency transport near the Nyquist limit and maintains sharply localized discontinuities in Burgers dynamics. The same construction extends to multidimensional problems and systems of conservation laws through oriented, vector-valued integer transfers. These results indicate that conservative dynamics admits an exact discrete realization in which both invariance and entropy selection are encoded at the operator level, rather than arising from approximate flux cancellation.