Nonlinear Time-Step Constraints Based on the Second Law of Thermodynamics

Abstract

Numerical calculations for a time-accurate solution of the equations of e uid dynamics often require a timestep 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 useof Fourier series to derive the CFL condition. In practice, computational e uid 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 e nally a system of equations representing the one-dimensional equations of gas dynamics. In each case results obtained agree with the classical approachforlinearequationsbutdifferinothers,indicatingthatthesecondlawhassignie cantimplicationsbeyond 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 hold for implicit and higher-order methods because of the universality of the second law and the concept of entropy.