Nonlinear time-step constraints based on the second law of thermodynamics

作     者：Camberos, JA 

作者机构：USAF Res Lab Wright Patterson AFB OH 45433 USA 

出 版 物：《JOURNAL OF THERMOPHYSICS AND HEAT TRANSFER》 (热物理学与热传导杂志)

年 卷 期：2000年第14卷第3期

页      面：435-449页

核心收录：

学科分类：07[理学] 0807[工学-动力工程及工程热物理] 0802[工学-机械工程] 070201[理学-理论物理] 0702[理学-物理学] 

主　　题：Second Law of Thermodynamics Entropy Generation Rate Lax Friedrichs Method Courant Friedrichs Lewy Rankine Hugoniot Jump Conditions Thermophysics Conservation of Mass Thermodynamic Equilibrium Nonlinear Equation Numerical Algorithms 

摘      要：Numerical calculations for a time-accurate solution of the equations of fluid dynamics often require a time-step constraint. One can reduce this constraint to an inequality relating the time step, the grid spacing, and some reference wave velocity. Historically, the literature in numerical analysis refers to this parametric cluster as the Courant number (nondimensional) and the condition for the linear case as the Courant-Friedrichs-Lewy (CFL) condition. Classically, numerical analysis relies on linearization and von Neumann s use of Fourier series to derive the CFL condition. In practice, computational fluid dynamics mostly relies on rules of thumb and heuristic arguments to justify the equation that determines time-step size and numerical stability for complicated and nonlinear calculations. The approach proposed in this paper uses the second law of thermodynamics as a way of imposing a restriction on the time step, applied to linear and nonlinear equations and systems of equations like the equations of gas dynamics. Basically, by transforming the truncation error for the numerical formula approximating a conservation equation into an equation representing the balance of entropy, one can obtain an inequality that restricts the time step to satisfy the second law. The second law as developed extends its role by analogy for the simple linear advection equation, then a nonlinear equation, and finally a system of equations representing the one-dimensional equations of gas dynamics. In each case results obtained agree with the classical approach for linear equations but differ in others, indicating that the second law has significant implications beyond its role in thermodynamics. This work develops the topic only for explicit numerical algorithms with truncation errors no greater than second order. By conjecture one expects that the most general conclusions will field for implicit and higher-order methods because of the universality of the second law and the concept o