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.

我们研究在一维空间中具有非凸通量的非线性双曲守恒律，对于基于零件算子求和的一类广泛的数值方法，我们用数值方法计算与每个方案相关的动力学函数。正如LeFloch和合作者所建立的那样，动力学函数（对于连续或离散模型）唯一地表征了小规模依赖，欠压缩，非经典冲击波的宏观动力学。我们在这里表明，各种熵耗散数值方案可以产生包含经典冲击的非经典解，包括具有（超）谱粘度的傅立叶方法，具有人工耗散的有限差分方案，具有或不具有模态滤波的不连续Galerkin方案以及TeCNO方案。我们用数值方法证明熵稳定性并不意味着一个空间维中标量守恒律的有限数值解的唯一性，并且我们计算了相关的动力学函数以区分这些方案。此外，我们为Keyfitz-Kranzer系统设计了耗散熵的方案，其解决方案是用三角波冲击进行测量。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有界。