发表机构
University of Trento; Southern University of Science and Technology; University of Vienna（特伦托大学; 南方科技大学; 维也纳大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出一类任意高阶精度、结构保持的时空格式，用于非线性声波系统，满足局部守恒、能量/熵稳定及旋度对合弱保持，并通过数学证明和数值测试验证。
AI 中文摘要
我们提出了一类新的结构保持、任意高阶精度的时空格式，用于求解一类非线性声波传播系统，该系统在空间上使用非结构化单纯形网格，其中非线性体现在状态变量与对偶变量之间的本构关系中，而微分算子保持标准的一阶声波结构。在全离散层面上，这些格式满足以下性质：I) 状态变量的局部守恒；II) 每个时空单元上的局部能量/熵平衡，具有一致的、单值的数值能量/熵通量和非负的局部能量/熵耗散，并且在省略稳定项时实现精确的能量守恒；III) 速度场旋度对合的弱保持，即使在存在间断且激活激波捕捉项的情况下也是如此。新格式的关键要素包括：i) 空间上的间断伽辽金（DG）离散化，其中对偶变量通过能量相对于守恒变量的导数在DG空间上的L2投影计算；ii) 数值通量和以对偶变量表示的、基于气泡加权的单元局部粘性正则化；iii) 时间上的连续-间断彼得罗夫-伽辽金方法。性质I)-III)首先在数学上得到证明，然后通过适当的数值测试进行验证。
英文摘要
We present a new class of structure-preserving, arbitrarily high-order accurate space-time schemes using unstructured simplicial meshes in space for a class of nonlinear acoustic wave propagation systems, in which the nonlinearity is encoded in the constitutive relations between the state and dual variables, while the differential operators retain the standard first-order acoustic wave structure. At the fully discrete level, the schemes satisfy the following properties: I) local conservation of the state variables; II) a local energy/entropy balance on each space-time element, with a consistent, single-valued numerical energy/entropy flux and nonnegative local energy/entropy dissipation, and exact energy conservation when the stabilization terms are omitted; III) weak preservation of the curl involution of the velocity field, even in the presence of discontinuities and with the shock-capturing terms activated. The key ingredients of the new schemes are: i) a discontinuous Galerkin (DG) discretization in space, in which the dual variables are computed as the L2 projections onto the DG space of the energy derivatives with respect to the conserved variables; ii) numerical fluxes and a bubble-weighted, element-local viscous regularization expressed in terms of the dual variables; iii) a continuous-discontinuous Petrov-Galerkin method in time. Properties I)-III) are first proved mathematically and are then verified by suitable numerical tests.
