Drift-Free Conservative Dynamics from Quantized Interaction Rules

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. 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. The results show that exact integer-transfer dynamics can suppress cumulative transport drift while preserving entropy-shock localization in nonlinear conservative evolution.