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

Numerical calculations for a time-accurate solution of the equations of flu id dynamics often require a time-step constraint. One can reduce this const raint to an inequality relating the time step, the grid spacing, and some r eference wave velocity. Historically, the literature in numerical analysis refers to this parametric cluster as the Courant number (nondimensional) an d the condition for the linear case as the Courant-Friedrichs-Lewy (CFL) co ndition. Classically, numerical analysis relies on linearization and von Ne umann's use of Fourier series to derive the CFL condition. In practice, com putational fluid dynamics mostly relies on rules of thumb and heuristic arg uments to justify the equation that determines time-step size and numerical stability for complicated and nonlinear calculations. The approach propose d 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 a nd systems of equations like the equations of gas dynamics. Basically, by t ransforming 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 s econd 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 dynam ics. In each case results obtained agree with the classical approach for li near equations but differ in others, indicating that the second law has sig nificant 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 th e universality of the second law and the concept of entropy.