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arXiv2026

A thermodynamically compatible semi-implicit finite volume scheme for the compressible Euler equations

In this work we construct a new semi-implicit (SI) hyperbolic and thermodynamically compatible (HTC) scheme for the compressible Euler equations of gasdynamics. The challenge therein lies in solving the entropy inequality as the primary evolution equation instead of the total energy conservation law, which in our framework is an additional conservation law that needs to be fulfilled by the numerical scheme as the consequence of a compatible discretization. The proposed method has a predictor-corrector character, where we first solve the isentropic problem without considering the entropy production. Therein, the entropy and the nonlinear convective terms in the momentum flux are discretized explicitly, while the mass flux and the pressure terms are taken implicitly. The discrete momentum equation is then inserted into the discrete mass conservation equation, leading to a mildly nonlinear system for the pressure, which can be efficiently solved with a nested Newton-type algorithm. Based on this prediction, a non-negative production term in the entropy is added, ensuring a cell entropy inequality for the numerical scheme. To guarantee global conservation of total energy at the discrete level, a new modified global Abgrall-type flux correction is introduced. The presented numerical method allows time steps that are restricted only by the material wave speed and not by the sound speed. This makes the scheme particularly useful for flows in the low-Mach-number regime. Numerical results are shown in one and two space dimensions to assess the theoretical properties of the new scheme.

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arXiv2026

A structure-preserving staggered semi-implicit four-split finite volume scheme for continuum mechanics on unstructured meshes

We present a new semi-implicit structure-preserving (SP) finite volume discretization for a unified first-order hyperbolic model of continuum mechanics. This framework provides a common mathematical description for fluids and solids and incorporates transport, viscous effects, heat conduction, and elastic deformations within a single system of hyperbolic partial differential equations. The coexistence of several physical mechanisms gives rise to multiple characteristic wave speeds, resulting in severe time-step restrictions for fully explicit discretizations. To overcome this difficulty, we develop a four-split scheme in which the governing equations are decomposed into convective, temperature, mechanical, and pressure subsystems. The convective subsystem is the only one that is advanced explicitly in time, while the remaining subsystems are treated implicitly in a sequential manner. As a consequence, the time step restriction of the proposed method depends only on the material velocity and is independent of the acoustic, shear, and thermal wave speeds that characterize the model. The scheme is designed to preserve key structural properties of the underlying equations on unstructured grids. In particular, a compatible vertex-staggered spatial discretization on triangles allows the curl-free involutions associated with the distortion field and thermal impulse to be respected whenever the relaxation source terms in the governing PDE are linear or absent. At the same time, the scheme is asymptotic preserving as it is consistent with the low Mach number limit and with the stiff relaxation limits of the GPR model that recover the classical Navier-Stokes-Fourier equations in fluid mechanics. A series of numerical experiments covering both fluids and solids demonstrates the accuracy, robustness, and multi-scale capabilities of the proposed approach.

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