A Legendre-pseudospectral method for computing travelling waves with corners (slope discontinuities) in one space dimension with application to Whitham’s equation family

Abstract

If the dispersion in a nonlinear hyperbolic wave equation is weak in the sense that the frequency ω( k) of cos( kx) is bounded as k→∞, it is common that (i) travelling waves exist up to a limiting amplitude with wave-breaking for higher amplitudes, and (ii) the limiting wave has a corner, that is, a discontinuity in slope. Because “corner” waves are not smooth, standard numerical methods converge poorly as the number of grid points is increased. However, the corner wave is important because, at least in some systems, it is the attractor for all large amplitude initial conditions. Here we devise a Legendre-pseudospectral method which is uncorrupted by the singularity. The symmetric ( u( X)= u(− X)) wave can be computed on an interval spanning only half the spatial period; since u is smooth on this domain which does not include the corner except as an endpoint, all numerical difficulties are avoided. A key step is to derive an extra boundary condition which uniquely identifies the corner wave. With both the grid point values of u( x) and phase speed c as unknowns, the discretized equations, imposing three boundary conditions on a second order differential equation, are solved by a Newton–Raphson iteration. Although our method is illustrated by the so-called “Whitham’s equation”, u t+uu x=∫ Du dx ′ where D is a very general linear operator, the ideas are widely applicable.