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In simulating physical systems, conservation of the total energy is oftenessential, especially when energy conversion between different forms of energyoccurs frequently. Recently, a new fourth order energy-preserving integratornamed MB4 was proposed based on the so-called continuous stage Runge--Kuttamethods (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 ofsecond order methods. In this paper, we illustrate how to apply the MB4 methodto a concrete ordinary differential equation using the nonlinearSchr\"{o}dinger-type equation on a two-dimensional grid as an example. Thissystem is a prototypical model of two-dimensional disordered organic materialand is difficult to solve with standard methods like the classical Runge--Kuttamethods due to the nonlinearity and the $\delta$-function like potential comingfrom defects. Numerical tests show that the method can solve the equationstably and preserves the total energy to 16-digit accuracy throughout thesimulation. It is also shown that parallelization of the method yields up to2.8 times speedup using 3 computational nodes.

A Parallelizable Energy-Preserving Integrator MB4 and Its Application to Quantum-Mechanical Wavepacket Dynamics