Entropy stable schemes replicate an entropy inequality at the semi-discrete level. These schemes rely on an algebraic summation-by-parts (SBP) structure and a technique referred to as flux differencing. We provide simple and efficient formulas for Jacobian matrices for the semi-discrete systems of ODEs produced by entropy stable discretizations. These formulas are derived based on the structure of flux differencing and derivatives of flux functions, which can be computed using automatic differentiation (AD). Numerical results demonstrate the efficiency and utility of these Jacobian formulas, which are then used in the context of two-derivative explicit time-stepping schemes and implicit time-stepping.

熵稳定方案在半离散水平上复制了熵不等式。这些方案依赖于分部代数求和 (SBP) 结构和称为通量差分的技术。我们为由熵稳定离散化产生的 ODE 的半离散系统提供了雅可比矩阵的简单有效的公式。这些公式是基于通量微分的结构和通量函数的导数推导出来的，可以使用自动微分 (AD) 进行计算。数值结果证明了这些雅可比公式的效率和实用性，然后将其用于二导显式时间步进方案和隐式时间步进的上下文中。