Overturning of nonlinear acoustic waves. Part 1 A general method

We consider model nonlinear wave equations of the form ut + uux = [Hscr ](x,t;u, ux,…) arising in gasdynamics and other fields, [Hscr ] incorporating various linear mechanisms of dissipation and dispersion. If [Hscr ] includes a thermoviscous dissipation term ∈uxx, then it is generally believed that u(x,t) will remain single-valued for all t > 0 and all single-valued u(x, 0), for any ∈ > 0. The question addressed here is whether, if thermoviscous dissipation is excluded from [Hscr ], u(x, t) remains single-valued for all t > 0, or whether certain dissipative-dispersive mechanisms (such as relaxation processes) are in themselves insufficient to prevent wave overturning. To answer this we propose a numerical scheme based on the use of intrinsic coordinates ψ = ψ(s, t) to describe the waveform at each time. In this paper, the method is described and validated by comparisons with the exact solutions for certain [Hscr ] ([Hscr ] = 0, [Hscr ] = -αu, [Hscr ] = ∈uxx). These comparisons show that the scheme is free of numerical viscosity effects which preclude the solution of the problem by finite-difference or spectral methods applied to the signal u(x, t), that it can reliably distinguish between finite-time overturning and merely the formation of steep gradients, and that it can accurately predict the time of overturning when it does occur. Having established the validity of the method, attention can then be turned to those cases where criteria for overturning have not as yet been determined by conventional methods. In Part 2, harmonic wave propagation through a relaxing gas is investigated.