We introduce a new family of high order accurate semi-implicit schemes for the solution of non-linear hyperbolic partial differential equations 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, 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 L2 projection operators from the FV to the VEM solution space and vice-versa. High order time accuracy is achieved using the semi-implicit IMEX Runge-Kutta schemes, and the novel schemes are proven to be asymptotic preserving and well-balanced. 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 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.

翻译：本文提出了一类新型高阶精确半隐式格式，用于求解非结构化多边形网格上的非线性双曲型偏微分方程。时间离散基于显式与隐式项的分裂，这种分裂可能源于控制方程的多尺度特性（涉及慢速与快速尺度），也可用于投影方法中（将数值解投影到物理意义解流形上）。我们提出对显式项采用高阶有限体积格式，以确保守恒性和对激波的鲁棒性；而虚拟元方法则用于处理通常需要求解椭圆问题的隐式项离散。通过合适的L2投影算子，数值解在有限体积与虚拟元解空间之间传输。半隐式IMEX龙格-库塔格式实现高阶时间精度，且新格式被证明具有渐近保持性和平衡性。以浅水方程和不可压缩纳维-斯托克斯方程为代表性模型：前者需处理由不同弗劳德数表征的多时间尺度，后者则借助投影方法求解以满足速度场无散约束。针对不可压缩纳维-斯托克斯模型，基于虚拟元技术设计了隐式黏性项离散。因此，最大允许时间步长的CFL型稳定性条件仅取决于流体速度，而非波速或黏性特征值。大量算例验证了该格式系列的精度及其在不可压缩流体领域解决相关基准问题的能力。