In simulating physical systems, conservation of the total energy is often essential, especially when energy conversion between different forms of energy occurs frequently. Recently, a new fourth order energy-preserving integrator named MB4 was proposed based on the so-called continuous stage Runge--Kutta methods (Y.~Miyatake and J.~C.~Butcher, SIAM J.~Numer.~Anal., 54(3), 1993-2013). A salient feature of this method is that it is parallelizable, which makes its computational time for one time step comparable to that of second order methods. In this paper, we illustrate how to apply the MB4 method to a concrete ordinary differential equation using the nonlinear Schrodinger-type equation on a two-dimensional grid as an example. This system is a prototypical model of two-dimensional disordered organic material and is difficult to solve with standard methods like the classical Runge--Kutta methods due to the nonlinearity and the $\delta$-function like potential coming from defects. Numerical tests show that the method can solve the equation stably and preserves the total energy to 16-digit accuracy throughout the simulation. It is also shown that parallelization of the method yields up to 2.8 times speedup using 3 computational nodes.

中文翻译：

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可并行化的保能积分器 MB4 及其在量子力学波包动力学中的应用

在模拟物理系统时，总能量的守恒往往是必不可少的，尤其是当不同形式的能量之间的能量转换频繁发生时。最近，基于所谓的连续阶段 Runge--Kutta 方法（Y.~Miyatake 和 J.~C.~Butcher，SIAM J.~Numer.~Anal. , 54(3), 1993-2013)。这种方法的一个显着特点是它是可并行化的，这使得其一个时间步的计算时间与二阶方法相当。在本文中，我们以二维网格上的非线性薛定谔型方程为例，说明如何将 MB4 方法应用于具体的常微分方程。该系统是二维无序有机材料的原型模型，由于非线性和来自缺陷的类似电位的 $\delta$ 函数，很难用经典的 Runge--Kutta 方法等标准方法求解。数值试验表明，该方法可以稳定求解方程，并在整个模拟过程中将总能量保持在16位精度。还表明，使用 3 个计算节点，该方法的并行化产生高达 2.8 倍的加速。