A robust numerical integrator for the short pulse equation near criticality
A robust numerical integrator for the short pulse equation near criticality
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DOI：
10.1016/j.cam.2019.04.002
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发表时间：
2019-12
期刊：
影响因子：
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通讯作者：
Shun Sato;K. Oguma;Takayasu Matsuo;B. Feng
中科院分区：
文献类型：
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作者：
Shun Sato;K. Oguma;Takayasu Matsuo;B. Feng
短脉冲方程由Schäfer-Wayne（2004）引入，用于模拟超短光脉冲的传播。虽然它可以描述广泛的解，但它的几个周期的超短脉冲解是传统非线性Schrödinger方程所不具备的，引起了人们的广泛关注。在这样的区域，解可能变得非常接近奇点，因此现有的数值方法不再稳定工作，或者即使它们稳定工作，也需要很高的计算成本。本文提出了一种鲁棒的数值积分方法，该方法是将一个全息变换与一些结构保持方法相结合而得到的。所得到的方案即使在奇点发生时也能成功地工作，从而为解决感兴趣的近奇点解提供了一种高度鲁棒的方法。数值实验证实了这一结论，并对导数爆破问题有了新的认识。
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, the solutions can become quite close to a singularity, and thus existing numerical methods cease to work stably, or even if they do, they require high computational cost. In this paper, we propose a robust numerical integration method which is obtained by combining a hodograph transformation and some structure-preserving methods. The resulting scheme successfully works even when the singularity occurs, and thus gives a highly robust method for resolving near singular solutions of interest. It is confirmed by numerical experiments, and some new insights about derivative blow-up are obtained.