We introduce Semi-Implicit Lagrangian Voronoi Approximation (SILVA), a novel numerical method for the solution of the incompressible Euler and Navier-Stokes equations, which combines the efficiency of semi-implicit time marching schemes with the robustness of time-dependent Voronoi tessellations. In SILVA, the numerical solution is stored at particles, which move with the fluid velocity and also play the role of the generators of the computational mesh. The Voronoi mesh is rapidly regenerated at each time step, allowing large deformations with topology changes. As opposed to the reconnection-based Arbitrary-Lagrangian-Eulerian schemes, we need no remapping stage. A semi-implicit scheme is devised in the context of moving Voronoi meshes to project the velocity field onto a divergence-free manifold. We validate SILVA by illustrative benchmarks, including viscous, inviscid, and multi-phase flows. Compared to its closest competitor, the Incompressible Smoothed Particle Hydrodynamics (ISPH) method, SILVA offers a sparser stiffness matrix and facilitates the implementation of no-slip and free-slip boundary conditions.

翻译：我们提出半隐式拉格朗日Voronoi近似（SILVA）——一种求解不可压缩Euler和Navier-Stokes方程的新型数值方法，该方法将半隐式时间推进格式的高效性与随时间变化的Voronoi剖分的鲁棒性相结合。在SILVA中，数值解存储于随流体速度运动的粒子上，这些粒子同时充当计算网格的生成元。每个时间步快速重构Voronoi网格，从而允许包含拓扑变化的大变形。与基于网格重构的任意拉格朗日-欧拉方法不同，本方法无需重映射步骤。针对移动Voronoi网格，我们设计了半隐式格式以将速度场投影至无散度流形。通过包含粘性流、无粘流及多相流的典型算例验证了SILVA方法的有效性。与最相近的竞争方法——不可压缩光滑粒子流体动力学（ISPH）方法相比，SILVA具有更稀疏的刚度矩阵，且便于实现无滑移和自由滑移边界条件。