A self-adaptive moving mesh method for the short pulse equation via its hodograph link to the sine-Gordon equation

Abstract

The short pulse equation was introduced by Schäfer- Wayne (2004) for modeling the propagation of ultrashort optical pulses. While it can describe a wide range of solutions, its ultrashort pulse solutions with a few cycles, which the conventional nonlinear Schrödinger equation does not possess, have drawn much attention. In such a region, existing numerical methods turn out to require very fine numerical mesh, and accordingly are computationally expensive. In this paper, we establish a new efficient numerical method by combining the idea of the hodograph transformation and the structure-preserving numerical methods. The resulting scheme is a self-adaptive moving mesh scheme that can successfully capture not only the ultrashort pulses but also exotic solutions such as loop soliton solutions.

On Spatial Discretization of Evolutionary Differential Equations on the Periodic Domain with a Mixed Derivative

“…Figure 1 shows that the average-difference is far better than the central difference, in particular, for high frequency components. This fact is in good agreement with the observation by Sato-Oguma-Matsuo-Feng [42] for the sine-Gordon equation (see, [42,Fig.14 and 16]). There, numerical solutions obtained by a finite difference method with the central difference strongly suffer from artificial oscillation, while those by the average-difference method reproduce the solution very well.…”

“…As a result of this exploration, the average-difference method, which has been recently introduced by the team including the present authors [20], turns out to be superior to other standard methods. This fact agrees very well with the numerical observation by Sato-Oguma-Matsuo-Feng [42] for the sine-Gordon equation. Summing up the findings above, we tentatively conclude that, for PDEs with a mixed derivative, the discretization of the differential form with the average-difference method is recommended.…”

“…As a result, we conclude that the average-difference is the best way to discretize the mixed derivative among 2nd order differences. This consequence agrees very well with the numerical observation for some specific cases [42,20].…”

“…Therefore, the solution u of (42) also satisfies (41) due to K k=1 fk (u) = 0, which is kept by the definition of C d . By using the theorem above, we see that the average-difference method [42] (43)…”