Fourier pseudospectral method with Kepler mapping for travelling waves with discontinuous slope: Application to corner waves of the Ostrovsky–Hunter equation and equatorial Kelvin waves in the four-mode approximation

Abstract

Many species of travelling waves have a single branch of solutions which ends at finite amplitude with a singular wave whose slope is discontinuous, a so-called “corner wave”. Fourier pseudospectral methods converge exponentially fast with N, the truncation of the series, for the smooth waves, but the error falls only as O(1/ N) for a wave with a slope discontinuity. We show that the error rate can be accelerated to O(1/ N 3) by making the change-of-coordinate (“Kepler map”) x = z − sin( z). Unfortunately, there is a subtlety: the end-in-a-singular-solution bifurcation is possible only for infinite-dimensional systems. We show that the truncated Fourier pseudospectral approximation has roots on both sides of the corner wave. The bifurcation can be detected by (i) a step-function-like jump in the residual of the differential equation at the corner wave and (ii) by observing the slope of u( x), which becomes discontinuous only at the corner wave itself. These concepts are illustrated for the equatorially trapped ocean Kelvin wave in the so-called “four-latitudinal-mode” approximation. However, corner waves arise in many species of waves and the concepts explained here are applicable to all.