In this article we present a novel staggered semi-implicit hybrid finite-volume/finite-element (FV/FE) method for the resolution of weakly compressible flows in two and three space dimensions. The pressure-based methodology introduced in [1], [2] for viscous incompressible flows is extended here to solve the compressible Navier-Stokes equations. Instead of considering the classical system including the energy conservation equation, we replace it by the pressure evolution equation written in non-conservative form. To ease the discretization of complex spatial domains, face-type unstructured staggered meshes are considered. A projection method allows the decoupling of the computation of the density and linear momentum variables from the pressure. Then, an explicit finite volume scheme is used for the resolution of the transport diffusion equations on the dual mesh, whereas the pressure system is solved implicitly by using continuous finite elements defined on the primal simplex mesh. Consequently, the CFL stability condition depends only on the flow velocity, avoiding the severe time restrictions that might be imposed by the sound velocity in the weakly compressible regime. High order of accuracy in space and time of the transport diffusion stage is attained using a local ADER (LADER) methodology. Moreover, also the CVC Kolgan-type second order in space and first order in time scheme is considered. To prevent spurious oscillations in the presence of shocks, an ENO-based reconstruction, the minmod limiter or the Barth-Jespersen limiter are employed. To show the validity and robustness of our novel staggered semi-implicit hybrid FV/FE scheme, several benchmarks are analysed, showing a good agreement with available exact solutions and numerical reference data from low Mach numbers, up to Mach numbers of the order of unity.

在本文中，我们提出了一种新颖的交错半隐式混合有限体积/有限元（FV / FE）方法，用于解决二维和三维空间中的弱可压缩流。在[1]，[2]中介绍的基于粘性的不可压缩流的基于压力的方法在此得到扩展，以求解可压缩的Navier-Stokes方程。与其考虑包括能量守恒方程在内的经典系统，不如将其替换为以非保守形式编写的压力演化方程。为了简化复杂空间域的离散化，考虑了面型非结构交错网格。投影方法允许将密度和线性动量变量的计算与压力解耦。然后，一个显式的有限体积方案用于解决双重网格上的输运扩散方程，而压力系统是通过使用在原始单纯形网格上定义的连续有限元隐式求解的。因此，CFL稳定性条件仅取决于流速，避免了在弱可压缩状态下声速可能施加的严格时间限制。使用局部ADER（LADER）方法可达到运输扩散级的空间和时间的高精确度。此外，还考虑了空间上的CVC Kolgan型二阶和时间方案的一阶。为了防止在出现电击时出现虚假振荡，采用了基于ENO的重建，minmod限制器或Barth-Jespersen限制器。