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

TL;DR

The work develops a structure-preserving, four-split semi-implicit finite-volume scheme for the GPR model, enabling unified treatment of fluids and solids while ensuring the timestep is governed by the material velocity rather than fast waves. By splitting into convective (explicit), and heat, G-J-v, and pressure (implicit) subsystems, the method achieves asymptotic preserving behavior in both low Mach and stiff relaxation limits, and exactly preserves curl involutions under suitable conditions. The approach employs a vertex-staggered grid with compatible discrete operators, enabling efficient linear solves (CG) for the implicit steps and a robust explicit flux for convection. Extensive numerical tests demonstrate accurate low-Mach incompressible limits, correct solid-fluid transitions, and curl-free preservation, underscoring the method’s potential for multi-physics continuum simulations. The paper also outlines future directions toward unstructured meshes, EM extensions, and higher-order spatial-temporal accuracy.

Abstract

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 within 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 the specific thermal impulse in the absence of source terms and were consistent 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 the finite, but potentially large, speed of heat waves. Instead, the new four-split semi-implicit scheme presented in this paper has a material time step restriction only. 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 three subsystems ii)-iv) are all discretized implicitly, hence a rather mild CFL restriction based on the velocity of the continuum is imposed. The method is asymptotically consistent with the low Mach number limit and the stiff relaxation limits. Moreover, it maintains an exactly curl-free distortion field and thermal impulse in the case of linear source terms or in their absence. The scheme is benchmarked against classical test cases verifying its theoretical properties.

Figures (13)

• Figure 1: Vertex-based staggered mesh, with the primary cells $\Omega_c$ and the dual cells $\Omega_p$.
• Figure 2: Numerical solution of the GPR model for the Taylor-Green vortex problem in the low Mach number limit and in the stiff relaxation limit with $\tau_1 = 10^{-8}$ and $\tau_2 = 10^{-10}$ at time $t=1.0$ using the new semi-implicit 4-split scheme. 1D cuts along the $x$ axis and comparison with the exact solution of the incompressible Navier-Stokes equations for the velocity components $u$ (left) and the pressure $p$ (right).
• Figure 3: Exact solution of the compressible Euler equations and numerical solution of the GPR model in the stiff relaxation limit ($\tau_1 = 10^{-10}, \tau_2 = 10^{-12}$) for Riemann problem RP1 (Sod shock tube) obtained with the new semi-implicit 4-split scheme. The density $\rho$, the velocity component $u$ and the pressure $p$ are shown at a final time of $t=0.2$.
• Figure 4: Exact solution of the Euler equations and numerical solution of the GPR model in the stiff relaxation limit ($\tau_1 = 10^{-10}, \tau_2 = 10^{-12}$) for Riemann problem RP2 (Lax shock tube). The density $\rho$, the velocity component $u$ and the pressure $p$ are shown at a final time of $t=0.14$.
• Figure 5: Reference solution and numerical solution of the homogeneous GPR model without source terms ($\tau_1 = \tau_2 = 10^{20}$) for Riemann problem RP3 at a final time of $t=0.2$. Top row: density $\rho$, the velocity component $v$ and the pressure $p$. Bottom row: distorsion field components $A_{11}$, $A_{21}$ and thermal impulse component $J_1$. One can note seven waves that are contained in the homogeneous part of the GPR model.