论文标题
非经典冲击，熵稳定性和分散求和的动力学功能
Kinetic functions for nonclassical shocks, entropy stability, and discrete summation by parts
论文作者
论文摘要
我们在一个空间维度上研究了具有非凸通量的非线性双曲保护定律，对于基于零件操作员的总和，我们在数值上计算了与每个方案相关的动力学函数。正如Lefloch和合作者确定的那样，动力学功能（对于连续或离散的模型）独特地表征了小规模依赖性，较底层，非分类的减震波的宏观尺度动力学。我们在这里表明，各种熵触点数值方案可以产生包含经典冲击的非经典溶液，包括具有（超）光谱粘度的傅立叶方法，具有人工耗散的有限差异方案，不连续的盖尔金方案，有或没有模态过滤，以及tecno schemes。我们从数值上证明熵稳定性并不意味着一个空间维度中标量保护定律的限制数值解的唯一性，并且我们计算了相关的动力学功能以区分这些方案。此外，我们为KeyFitz-Kranzer系统设计熵 - 触及式方案，其解决方案是带有三角冲击的度量。该系统说明了一个事实，即熵稳定性并不意味着在网格细化下的界限。
We study nonlinear hyperbolic conservation laws with non-convex flux in one space dimension and, for a broad class of numerical methods based on summation by parts operators, we compute numerically the kinetic functions associated with each scheme. As established by LeFloch and collaborators, kinetic functions (for continuous or discrete models) uniquely characterize the macro-scale dynamics of small-scale dependent, undercompressive, nonclassical shock waves. We show here that various entropy-dissipative numerical schemes can yield nonclassical solutions containing classical shocks, including Fourier methods with (super-) spectral viscosity, finite difference schemes with artificial dissipation, discontinuous Galerkin schemes with or without modal filtering, and TeCNO schemes. We demonstrate numerically that entropy stability does not imply uniqueness of the limiting numerical solutions for scalar conservation laws in one space dimension, and we compute the associated kinetic functions in order to distinguish between these schemes. In addition, we design entropy-dissipative schemes for the Keyfitz-Kranzer system whose solutions are measures with delta shocks. This system illustrates the fact that entropy stability does not imply boundedness under grid refinement.
