发表机构
University of Trento; Southern University of Science and Technology; Université Savoie Mont Blanc; University of Ferrara; University of Verona; Inria; Université de Strasbourg（特伦托大学; 南方科技大学; 萨瓦-蒙布朗大学; 费拉拉大学; 维罗纳大学; 法国国家数字研究院; 斯特拉斯堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对连续介质力学的统一双曲模型，提出一种四分裂半隐式有限体积格式，通过仅显式处理对流项来放宽时间步长限制，并在非结构网格上保持结构性质，数值实验验证了其精度与多尺度能力。
AI 中文摘要
我们提出了一种新的半隐式保持结构（SP）有限体积离散方法，用于统一的一阶双曲型连续介质力学模型。该框架为流体和固体提供了共同的数学描述，并在单一的双曲型偏微分方程系统中包含了输运、粘性效应、热传导和弹性变形。多种物理机制的共存产生了多个特征波速，导致全显式离散化面临严格的时间步长限制。为克服这一困难，我们开发了一种四分裂格式，将控制方程分解为对流、温度、力学和压力子系统。对流子系统是唯一在时间上显式推进的子系统，而其余子系统则按顺序隐式处理。因此，所提方法的时间步长限制仅取决于物质速度，而与表征该模型的声速、剪切波速和热波速无关。该格式旨在非结构网格上保持底层方程的关键结构性质。特别是，在三角形网格上采用相容的顶点交错空间离散化，使得当控制偏微分方程中的松弛源项为线性或不存在时，与畸变场和热冲量相关的无旋对合（curl-free involutions）得以保持。同时，该格式具有渐近保持性，与低马赫数极限以及GPR模型的刚性松弛极限一致，后者可恢复流体力学中的经典纳维-斯托克斯-傅里叶方程。一系列涵盖流体和固体的数值实验证明了所提方法的准确性、鲁棒性和多尺度能力。
英文摘要
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.
