Park Junhu | Searcharxiv

arXiv2026

Drift-Free Conservative Dynamics from Quantized Interaction Rules

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.

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arXiv2026

Continuum dynamics from quantised interaction rules

Hyperbolic conservation laws are conventionally solved by evolving reconstructed floating-point fields, incurring both computational overhead and structural diffusion near discontinuities. Here we introduce the Fast Quantised Numerical Method (FQNM), in which the conservative operator is realised directly as an antisymmetric integer transfer rule on a countable state space, with continuum fields appearing only as reconstructed observables. For scalar conservation laws with monotone flux splitting, we establish exact conservation, monotonicity, TVD and $L^1$ stability, and convergence of the reconstructed solution to the entropy solution under $\delta/\Delta x \to 0$. We further show that distinct classical flux formulations collapse to identical dynamics whenever they induce the same integer transfer rule, identifying the transfer operator as the effective computational object. Across representative regimes, FQNM remains stable near the Nyquist limit in high-frequency transport, preserves grid-level shock structure in Burgers dynamics, and in a matched Roe-flux Sod prototype preserves shock structure at the density-scale conserved-state level relative to an exact Riemann reference, while achieving order-of-magnitude prototype acceleration over floating-point baselines. These results demonstrate that, for conservative hyperbolic dynamics, executing the operator as quantised transfer rather than reconstructed field evolution can simultaneously alter structural fidelity and reduce computational cost, establishing a new representation paradigm for conservation-law solvers.

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