基于Pade逼近的Rosenau-Hyman双曲子的稳定性和动力学性质
Stability and dynamical properties of Rosenau-Hyman compactons using Pade approximants
摘要：
我们提出了一个系统的方法计算高阶导数的光滑函数在均匀网格上使用Padé逼近。我们说明了我们的研究结果，通过推导高阶近似使用传统的二阶有限差分公式作为我们的出发点。我们利用这些方案研究了K（2，2）Rosenau-Hyman（RH）介子的稳定性和动力学性质，包括两个介子的碰撞和由此产生的激波的形成。我们的方法使用的差分方案，只涉及最近的和下一个最近的邻居在一个统一的空间网格。的偏微分方程的牛顿涉及的第一，第二和第三偏导数的空间坐标，我们集中在四个不同的四阶方法不同的可能性增加的准确度（或不）的空间导数之一，六阶。在单孤立波稳定性研究中，一种减小舍入误差的方法被认为是最精确的近似方法，尽管所有的导数都只精确到四阶。模拟牛顿散射需要添加与人工粘性相关的四阶导数。对于这些问题，不同的选择会导致不同数量的“杂散”辐射，我们比较了不同选择的优点。
英文摘要：
We present a systematic approach for calculating higher-order derivatives of smooth functions on a uniform grid using Padé approximants. We illustrate our findings by deriving higher-order approximations using traditional second-order finite-differences formulas as our starting point. We employ these schemes to study the stability and dynamical properties of K(2,2) Rosenau-Hyman (RH) compactons including the collision of two compactons and resultant shock formation. Our approach uses a differencing scheme involving only nearest and next-to-nearest neighbors on a uniform spatial grid. The partial differential equation for the compactons involves first, second and third partial derivatives in the spatial coordinate and we concentrate on four different fourth-order methods which differ in the possibility of increasing the degree of accuracy (or not) of one of the spatial derivatives to sixth order. A method designed to reduce roundoff errors was found to be the most accurate approximation in stability studies of single solitary waves, even though all derivates are accurate only to fourth order. Simulating compacton scattering requires the addition of fourth derivatives related to artificial viscosity. For those problems the different choices lead to different amounts of "spurious" radiation and we compare the virtues of the different choices.
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