一种新的NS方程算法执行过程的数学框架

A New Mathematical Framework for Describing the Execution Process of NS Equation Algorithms

摘要: Navier-Stokes (NS)方程的数值求解依赖算子分裂、多物理场耦合等复杂算法，但传统数学工具(如集合论、范畴论)难以严格描述算法选择逻辑与执行过程：集合论无法刻画动态操作–结果关联，范畴论虽具备抽象统一视角却缺乏计算可行性，导致算法设计常依赖经验，执行过程“黑箱化”。为解决此问题，本文提出基于广义映射理论(GMT)的数学框架，通过“对象集–操作集–结果集–生成关系”四元结构，将NS方程数值求解的静态参数(物理量、网格)、动态操作(对流/粘性离散、压力重构)、分支结果(多算法选择)转化为可计算的张量映射。该框架不仅像范畴论一样为算法提供严格数学语言，更通过操作显式化、路径可追踪特性，使算子分裂的子步骤、多模型融合的概率权重均成为透明可控的映射单元。以汽车外流场模拟、血栓形成血流模拟为案例，验证框架可精准描述不同NS算法的执行逻辑，且支持流场结果与计算性能的定量追踪。实验表明，相比传统描述方法，该框架使算法选择的数学一致性和执行过程的可解释性的实现成为可能，为NS方程数值算法的标准化、透明化提供新工具。

Abstract: The numerical solution of the Navier-Stokes (NS) equations relies on complex algorithms such as operator splitting and multi-physics field coupling. However, traditional mathematical tools (e.g., set theory, category theory) struggle to rigorously describe the algorithm selection logic and execution process: set theory cannot characterize the dynamic operation-result relationships, while category theory, despite its abstract and unified perspective, lacks computational feasibility. This leads to algorithm design often depending on experience, resulting in an “black-box” execution process. To address this issue, this paper proposes a mathematical framework based on the Generalized Mapping Theory (GMT). Through a four-element structure of “object set-operation set-result set-generation relationship”, it transforms the static parameters (physical quantities, grids), dynamic operations (convection/viscous discretization, pressure reconstruction), and branch results (multiple algorithm selections) in the numerical solution of NS equations into computable tensor mappings. This framework not only provides a rigorous mathematical language for algorithms like category theory but also, through the explicit nature of operations and traceable path characteristics, makes sub-steps of operator splitting and probability weights of multi-model fusion transparent and controllable mapping units. Taking automotive external flow field simulation and thrombus formation blood flow simulation as cases, it is verified that the framework can accurately describe the execution logic of different NS algorithms and support the quantitative tracking of flow field results and computational performance. Experiments show that compared with traditional description methods, this framework enables the realization of mathematical consistency in algorithm selection and interpretability in the execution process, providing a new tool for the standardization and transparency of NS equation numerical algorithms.