We propose a new pressure-based structure-preserving (SP) and quasi asymptotic preserving (AP) staggered semi-implicit finite volume scheme for the unified first order hyperbolic formulation of continuum mechanics [1], which goes back to the pioneering work of Godunov [2] and further work of Godunov & Romenski [3] and Peshkov & Romenski [4]. The unified model is based on the theory of symmetric-hyperbolic and thermodynamically compatible (SHTC) systems [2], [5] and includes the description of elastic and elasto-plastic solids in the nonlinear large-strain regime as well as viscous and inviscid heat-conducting fluids, which correspond to the stiff relaxation limit of the model. In the absence of relaxation source terms, the homogeneous PDE system is endowed with two stationary linear differential constraints (involutions), which require the curl of distortion field and the curl of the thermal impulse to be zero for all times. In the stiff relaxation limit, the unified model tends asymptotically to the compressible Navier-Stokes equations.

The new structure-preserving scheme presented in this paper can be proven to be exactly curl-free for the homogeneous part of the PDE system, i.e. in the absence of relaxation source terms. We furthermore prove that the scheme is quasi asymptotic preserving in the stiff relaxation limit, in the sense that the numerical scheme reduces to a consistent second order accurate discretization of the compressible Navier-Stokes equations when the relaxation times tend to zero. Last but not least, the proposed scheme is suitable for the simulation of all Mach number flows thanks to its conservative formulation and the implicit discretization of the pressure terms.

我们为连续统力学的统一一阶双曲公式提出了一种新的基于压力的结构保持（SP）和拟渐近保存（AP）交错半隐式有限体积方案[1]，这可追溯到Godunov的开创性工作。 [2]和Godunov＆Romenski [3]和Peshkov＆Romenski [4]的进一步工作。统一模型基于对称双曲线和热力学兼容（SHTC）系统的理论[2]，[5]，包括非线性大应变状态下的弹性和弹塑性固体以及粘性和无粘性的描述。导热流体，它对应于模型的刚性松弛极限。在没有放松源条款的情况下，对合），要求变形场的弯曲度和热冲击的弯曲度始终为零。在刚性松弛极限下，统一模型趋近于可压缩的Navier-Stokes方程。

可以证明本文提出的新的结构保留方案对于PDE系统的同质部分是完全无卷曲的，即在没有松弛源项的情况下。我们进一步证明，在松弛时间趋于零时，数值方案将可压缩的Navier-Stokes方程简化为一致的二阶精确离散化，从这个意义上讲，该方案在刚性松弛极限下是拟渐近的。最后但并非最不重要的一点是，由于其保守的公式表示法和压力项的隐式离散，所提出的方案适用于所有马赫数流的仿真。