The discrete ordinate/pseudo-spectral method: review and application from a physicist's perspective
Australian Journal of Physics
Peakons and coshoidal waves: traveling wave solutions of the Camassa-Holm equation
Applied Mathematics and Computation
A MATLAB differentiation matrix suite
ACM Transactions on Mathematical Software (TOMS)
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
The cnoidal wave/corner wave/breaking wave scenario: A one-sided infinite-dimension bifurcation
Mathematics and Computers in Simulation - Special issue: Nonlinear waves: Computation and theory III
Journal of Computational Physics
Hi-index | 31.45 |
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", ut + uux = ∫ Du dx' where D is a very general linear operator, the ideas are widely applicable.