表格状态方程的热力学导数的一致估计
Consistent thermodynamic derivative estimates for tabular equations of state
摘要：
可压缩流体流动的数值模拟需要状态方程（EOS）来关联密度、内能、温度和压力的热力学变量。一个有效的状态方程必须满足一致性（从自由能导出）和稳定性（正声速平方）的热力学条件。当相变显著时，EOS是复杂的并且只能在表中指定。对于表格式的EOS，例如来自洛斯阿拉莫斯国家实验室的SEEOS，一致性和稳定性条件采取微分方程的形式，该微分方程将压力和能量的导数与正性约束一起作为温度和密度的函数。基于多项式或有理插值的与这种表的典型软件接口计算压力和能量的导数，并且可以强制稳定性条件，但不强制一致性条件及其导数。我们描述了一种新类型的表接口的基础上约束局部最小二乘回归技术。它被应用到几个SEEST-EOS的显示如何的一致性条件可以满足四舍五入，而计算一阶和二阶导数证明二阶收敛。一个14个数量级的改进比传统的衍生物证明，虽然新的方法显然是两个数量级慢，由于这样的事实，即每个评估需要解决一个11维的非线性系统。
英文摘要：
Numerical simulations of compressible fluid flows require an equation of state (EOS) to relate the thermodynamic variables of density, internal energy, temperature, and pressure. A valid EOS must satisfy the thermodynamic conditions of consistency (derivation from a free energy) and stability (positive sound speed squared). When phase transitions are significant, the EOS is complicated and can only be specified in a table. For tabular EOS's such as SESAME from Los Alamos National Laboratory, the consistency and stability conditions take the form of a differential equation relating the derivatives of pressure and energy as functions of temperature and density, along with positivity constraints. Typical software interfaces to such tables based on polynomial or rational interpolants compute derivatives of pressure and energy and may enforce the stability conditions, but do not enforce the consistency condition and its derivatives. We describe a new type of table interface based on a constrained local least squares regression technique. It is applied to several SESAME EOS's showing how the consistency condition can be satisfied to round-off while computing first and second derivatives with demonstrated second-order convergence. An improvement of 14 orders of magnitude over conventional derivatives is demonstrated, although the new method is apparently two orders of magnitude slower, due to the fact that every evaluation requires solving an 11-dimensional nonlinear system.
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