Fourth Order Time-Stepping for Kadomtsev–Petviashvili and Davey–Stewartson Equations

Abstract

Abstract. Purely dispersive partial differential equations as the Korteweg-de Vries equation, the nonlinear Schrödinger equation and higher dimensional generalizations thereof can have solutions which develop a zone of rapid modulated oscillations in the region where the corresponding dispersionless equations have shocks or blow-up. To numerically study such phenomena, fourth order time-stepping in combination with spectral methods is beneficial to resolve the steep gradients in the oscillatory region. We compare the performance of several fourth order methods for the KadomtsevPetviashvili and the Davey-Stewartson equations, two integrable equations in 2+1 dimensions: exponential time-differencing, integrating factors, time-splitting, implicit Runge-Kutta and Driscoll's composite Runge-Kutta method. The accuracy in the numerical conservation of integrals of motion is discussed.

Spectral Approach to D‐bar Problems

“…The approach is then applied to the D-bar problem of the Davey-Stewartson (DS) II equation, and the resulting solution is compared to a direct numerical solution of DS II as in [13].…”

“…In [13] it was shown that the error in the numerical conservation of the DS II energy tends to overestimate the numerical accuracy of the solution in an L ∞ sense by two to three orders of magnitude.…”

A numerical approach to blow-up issues for dispersive perturbations of Burgers’ equation

“…For equations of the form (23) with diagonal L, there are many efficient high-order time integrators, see e.g. [30,23,33,35] and references therein. This allows for an efficient integration in time.…”

“…Accuracy of the numerical solution is controlled as discussed in [33,35,34] via the numerically computed energies (10) and (14) which will depend on time due to unavoidable numerical errors. We use the quantity…”

Some special solutions to the Hyperbolic NLS equation

“…The step size is chosen adaptively using the socalled H211b digital filter [56,57] to meet the prescribed error tolerance, set as of the order of machine precision. A more detailed study of time-discretization accuracy for similar spectral models can be found in [38].…”