Continuum dynamics from quantised interaction rules
This addresses numerical conservation issues in computational physics simulations, offering a novel approach that is incremental but with specific improvements over existing methods.
The paper tackles the problem of numerical conservation in continuum dynamics by formulating conservative evolution directly as quantised interaction rules on countable states, resulting in the Fast Quantised Numerical Method (FQNM) that maintains accuracy in high-frequency transport and preserves grid-level structure in nonlinear shock formation.
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.