We introduce a new family of high order accurate semi-implicit schemes for the solution of nonlinear time-dependent systems of partial differential equations (PDE) on unstructured polygonal meshes. The time discretization is based on a splitting between explicit and implicit terms that may arise either from the multi-scale nature of the governing equations, which involve both slow and fast scales, or in the context of projection methods, where the numerical solution is projected onto the physically meaningful solution manifold. We propose to use a high order finite volume (FV) scheme for the explicit terms, hence ensuring conservation property and robustness across shock waves, while the virtual element method (VEM) is employed to deal with the discretization of the implicit terms, which typically requires an elliptic problem to be solved. The numerical solution is then transferred via suitable L${}_2$ projection operators from the FV to the VEM solution space and vice-versa. High order time accuracy is then achieved using the semi-implicit IMEX Runge–Kutta schemes, and the novel schemes are proven to be asymptotic preserving (AP) and well-balanced (WB). As representative models, we choose the shallow water equations (SWE), thus handling multiple time scales characterized by a different Froude number, and the incompressible Navier–Stokes equations (INS), which are solved at the aid of a projection method to satisfy the solenoidal constraint of the velocity field. Furthermore, an implicit discretization for the viscous terms is also devised for the INS model, which is based on the VEM technique. Consequently, the CFL-type stability condition on the maximum admissible time step is based only on the fluid velocity and not on the celerity nor on the viscous eigenvalues. A large suite of test cases demonstrates the accuracy and the capabilities of the new family of schemes to solve relevant benchmarks in the field of incompressible fluids.

我们引入了一系列新的高阶精确半隐式方案，用于求解非结构化多边形网格上的非线性瞬态偏微分方程 (PDE)系统. 时间离散化基于显式和隐式项之间的拆分，这可能是由于控制方程的多尺度性质（涉及慢速和快速尺度），或者在投影方法的背景下产生的，其中数值解被投影到具有物理意义的解流形上。我们建议对显式项使用高阶有限体积 (FV) 方案，从而确保冲击波的守恒性和鲁棒性，同时采用虚拟元方法 (VEM) 来处理隐式项的离散化，这通常需要解决一个椭圆问题。然后通过合适的 L 传递数值解${}_{\mathrm{2个}}$从 FV 到 VEM 解空间的投影算子，反之亦然。然后使用半隐式 IMEX Runge–Kutta 方案实现了高阶时间精度，并且新方案被证明是渐近保持 (AP) 和良好平衡 (WB) 的。作为代表性模型，我们选择浅水方程（SWE），从而处理以不同弗劳德数为特征的多个时间尺度，以及不可压缩的 Navier-Stokes 方程（INS），这些方程在投影方法的帮助下求解以满足速度场的螺线管约束. 此外，还为基于 VEM 技术的 INS 模型设计了粘性项的隐式离散化。因此，在最大允许时间步长上的 CFL 型稳定性条件仅基于流体速度而不是速度或粘性特征值。大量测试用例展示了新系列方案解决不可压缩流体领域相关基准的准确性和能力。