A structure-preserving semi-implicit four-split scheme for continuum mechanics

In this work we introduce a novel structure-preserving vertex-staggered semi-implicit four-split discretization of a unified first order hyperbolic formulation of continuum mechanics that is able to describe at the same time fluid and solid materials in one and the same mathematical model. The governing PDE system goes back to pioneering work of Godunov, Romenski, Peshkov and collaborators. Previous structure-preserving discretizations of this system allowed to respect the curl-free properties of the distortion field and of the specific thermal impulse in the absence of source terms and were also able to properly deal with the low Mach number limit with respect to the adiabatic sound speed. However, the evolution of the thermal impulse and the distortion field were still discretized explicitly, thus requiring a rather severe CFL stability restriction on the time step based on the shear sound speed and on the finite, but potentially large, speed of heat waves. Instead, the new four-split semi-implicit scheme presented in this paper has a CFL time step restriction based only on the magnitude of the velocity field of the continuum. For this purpose, the governing PDE system is split into four subsystems: i) a convective subsystem, which is the only one that is treated explicitly; ii) a heat subsystem, iii) a subsystem containing momentum, distortion field and specific thermal impulse; iv) a pressure subsystem. The last three subsystems ii)-iv) are all discretized implicitly, hence the time step is only limited by a rather mild CFL condition based on the magnitude of the velocity field. The method is consistent with the low Mach number limit of the equations, with the stiff relaxation limits and it maintains an exactly curl-free distortion field and thermal impulse in the case of linear source terms or in their absence. We show several numerical results for classical benchmark problems that allow to assess the performance of the scheme in different asymptotic limits of the governing equations, including the fluid and solid limit.