Conservation Framework and 2D Advection-Diffusion Simulation Universe OS is a cross-scale, strong-conservation, and high-performance numerical simulation framework designed to address long-standing challenges in traditional computational science, including poor numerical stability, lack of energy conservation, and fragmentation of multi-domain solvers. Through a unified mathematical foundation

Abstract

Universe OS is a cross-scale, strong-conservation, and high-performance numerical simulation framework designed to address long-standing challenges in traditional computational science, including poor numerical stability, lack of energy conservation, and fragmentation of multi-domain solvers. Through a unified mathematical foundation and modular architecture, it transforms scattered disciplinary fields into pluggable application modules, enabling self-consistent full-scale simulation from microscopic particles to macroscopic fields. As a core application module of the framework, the 2D advection-diffusion numerical simulation provides a complete implementation from basic functions to advanced extensions, centered around the AdvectionDiffusion2D class. The core advantage of the framework stems from its self-consistent mathematical system and forced conservation design, fundamentally eliminating the accumulation of numerical errors. Its five-dimensional unified evolution operator integrates spatial field quantities, energy, time, mass transport, and microscopic interactions into a single computational loop, achieving mathematical unification of all physical processes. Simultaneously, it introduces an energy account E_{\text{heat}} (tracking numerical errors) and a reality correction Z_{\text{proxy}} (feeding errors back to the system) mechanism, ensuring absolute conservation of total energy in each simulation step and completely solving the problem of numerical dissipation in traditional methods. At the level of micro-macro coupling, the framework converts complex particle behaviors (such as injection \eta_{\text{in}} and fusion \eta_{\text{fus}}) into net contributions to the macroscopic field through the microscopic agent N_{\text{expect}} and Monte Carlo sampling, avoiding the high cost of traditional molecular dynamics/quantum simulations. Combined with high-dimensional composite operators, it abstracts microscopic interactions into simplified conservative composite operators, achieving a leap in cross-scale computing performance. The AdvectionDiffusion2D class, a core application of the framework, boasts rich functional features and flexible boundary condition support. It supports adaptive time