We present a particle-grid characteristic-mapping framework that extends long-range characteristic mapping from inviscid flows to general Navier-Stokes dynamics with viscosity, body forces, and complex boundaries. Unlike traditional grid-based and vorticity-centered characteristic methods, our method is built on the observation that particle trajectories naturally provide the long-range flow map, enabling geometric quantities and their gradients to be transported in a direct and effective manner. We identify the impulse, the gauge variable of the velocity field, as the primary quantity mapped along characteristics while remaining compatible with standard velocity-based incompressible solvers. Using the 1-form representation of the impulse equation, we derive an integral formulation that decomposes the impulse evolution into a component transported geometrically along the particle flow map and a complementary component generated by viscosity and body forces evaluated through path integrals accumulated along particle trajectories. These components together yield a unified characteristic-mapping solver capable of handling incompressible Navier-Stokes flows with viscosity and body forces while maintaining the accuracy and geometric fidelity of characteristic transport.

翻译：本文提出了一种粒子-网格特征映射框架，将长程特征映射从无粘流动推广至包含粘性、体积力及复杂边界的一般Navier-Stokes动力学。与传统基于网格和以涡量为核心的特征线方法不同，我们的方法基于以下观察：粒子轨迹天然提供了长程流映射，使得几何量及其梯度能够以直接有效的方式被输运。我们选取冲量——速度场的规范变量——作为沿特征线映射的主要物理量，同时保持与标准基于速度的不可压缩求解器的兼容性。利用冲量方程的1-形式表示，我们推导出积分形式，将冲量演化分解为沿粒子流映射几何输运的分量，以及由粘性与体积力通过沿粒子轨迹累积的路径积分所生成的分量。这些分量共同构成了统一的特征映射求解器，能够处理包含粘性和体积力的不可压缩Navier-Stokes流动，同时保持特征输运的精度与几何保真性。