论文标题
补充可变方法，用于开发结构的数值近似值，以热力学上一致的部分微分方程
Supplementary Variable Method for Developing Structure-Preserving Numerical Approximations to Thermodynamically Consistent Partial Differential Equations
论文作者
论文摘要
我们提出了一种新的时间离散范式，用于将能源生产率保留数值近似值开发到热力学上一致的部分微分方程系统，称为补充变量方法。其背后的核心思想是向热力学一致的模型引入补充变量，以使过度确定的方程式系统由热力学一致的PDE系统组成，能量定义和能量耗散方程，结构稳定。补充变量允许人们在时间离散后保留能量耗散方程与PDE系统之间的一致性。我们使用耗散梯度流模型说明了该方法。在几乎无限的许多可能性中，我们提出了两种在梯度流模型中添加补充变量的方法，以开发能量降低速率保存算法。使用伪谱法进行空间离散化。然后，我们将两个新方案与能量稳定的SAV方案和完全隐式的曲柄 - 尼科尔森方案进行比较。结果有利于整体绩效的新计划。这种新的数值范式可以应用于任何热力学一致的模型。
We present a new temporal discretization paradigm for developing energy-production-rate preserving numerical approximations to thermodynamically consistent partial differential equation systems, called the supplementary variable method. The central idea behind it is to introduce a supplementary variable to the thermodynamically consistent model to make the over-determined equation system, consisting of the thermodynamically consistent PDE system, the energy definition and the energy dissipation equation, structurally stable. The supplementary variable allows one to retain the consistency between the energy dissipation equation and the PDE system after the temporal discretization. We illustrate the method using a dissipative gradient flow model. Among virtually infinite many possibilities, we present two ways to add the supplementary variable in the gradient flow model to develop energy-dissipation-rate preserving algorithms. Spatial discretizations are carried out using the pseudo-spectral method. We then compare the two new schemes with the energy stable SAV scheme and the fully implicit Crank-Nicolson scheme. The results favor the new schemes in the overall performance. This new numerical paradigm can be applied to any thermodynamically consistent models.
