Park Junhu

Park Junhu, Youngsoo Ha, Myungjoo Kang

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.

Park Junhu, Yongsoo Ha, Myungjoo Kang

Conservative dynamics are typically computed as floating-point approximations to continuum differential operators, which can obscure conservation through rounding and discretisation artefacts. Here we instead formulate conservative evolution directly as quantised interaction rules acting on countable states. The resulting Fast Quantised Numerical Method (FQNM) executes dynamics through antisymmetric integer transfer, with physical fields appearing only after reconstruction. In high-frequency transport, the method remains accurate deep into the Nyquist regime where a standard high-order floating-point baseline deteriorates. In nonlinear shock formation, it preserves grid-level structure and remains robust to cell drifting while maintaining exact discrete conservation. These results show that conservative dynamics can be executed directly through discrete interaction rules, with continuum behaviour emerging only as a reconstruction of underlying quantised states.