A parallelizable energy-preserving integrator MB4 and its application to quantum-mechanical wavepacket dynamics. (English) Zbl 1471.65079

Summary: 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, No. 3, 1993–2013 (2016; Zbl 1342.65232)]. 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 Schrödinger-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.


65L05 Numerical methods for initial value problems involving ordinary differential equations
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65P10 Numerical methods for Hamiltonian systems including symplectic integrators
65Y05 Parallel numerical computation
68W10 Parallel algorithms in computer science


Zbl 1342.65232


mb4-nls2d; GitHub
Full Text: DOI arXiv


[1] Hairer, E.; Lubich, C.; Wanner, G., Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations (2006), Berlin: Springer, Berlin · Zbl 1094.65125
[2] Gonzalez, O., Time integration and discrete Hamiltonian systems, J. Nonlinear Sci., 6, 449-467 (1996) · Zbl 0866.58030
[3] McLachlan, RI; Quispel, GRW; Robidoux, N., Geometric integration using discrete gradients, Phil. Trans. R. Soc. Lond. A, 357, 1021-1045 (1999) · Zbl 0933.65143
[4] Quispel, GRW; McLaren, DI, A new class of energy-preserving numerical integration methods, J. Phys. A, 41, 045206 (2008) · Zbl 1132.65065
[5] Hairer, E., Energy-preserving variant of collocation methods, J. Numer. Anal. Ind. Appl. Math., 5, 73-84 (2010) · Zbl 1432.65185
[6] Miyatake, Y.; Butcher, JC, A characterization of energy-preserving methods and the construction of parallel integrators for Hamiltonian systems, SIAM J. Numer. Anal., 54, 1993-2013 (2016) · Zbl 1342.65232
[7] Delfour, M.; Fortin, M.; Payr, G., Finite-difference solutions of a non-linear Schrödinger equation, J. Comput. Phys., 44, 277-288 (1981) · Zbl 0477.65086
[8] Besse, C., A relaxation scheme for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 42, 934-952 (2004) · Zbl 1077.65103
[9] Sanz-Serna, JM; Verwer, JG, Conerservative and nonconservative schemes for the solution of the nonlinear Schrödinger equation, IMA J. Numer. Anal., 6, 25-42 (1986) · Zbl 0593.65087
[10] GitHub. (2019). https://github.com/mb4-nls2d/mb4-nls2d/tree/master/output/
[11] GitHub. (2019). https://github.com/mb4-nls2d/
[12] Hammock, ML; Chortos, A.; Tee, BC-K; Tok, JB-H; Bao, Z., 25th Anniversary article: the evolution of electronic skin (e-skin): a brief history, design considerations, and recent progress, Adv. Mater., 25, 5997-6038 (2013)
[13] Troisi, A.; Orlandi, G., Charge-transport regime of crystalline organic semiconductors: diffusion limited by thermal off-diagonal electronic disorder, Phys. Rev. Lett., 96, 086601/1-4 (2006)
[14] Imachi, H.; Yokoyama, S.; Kaji, T.; Abe, Y.; Tada, T.; Hoshi, T., One-hundred-nm-scale electronic structure and transport calculations of organic polymers on the K computer, AIP Conf. Proc., 1790, 020010/1-4 (2016)
[15] Hoshi, T., Imachi, H., Kumahata, K., Terai, M., Miyamoto, K., Minami, K., Shoji, F.: Extremely scalable algorithm for \(10^8\)-atom quantum material simulation on the full system of the K computer. In: Proceedings of ScalA16: workshop on latest advances in scalable algorithms for large-scale systems, held in conjunction with SC16: the international conference on high performance computing, networking, storage and analysis, Salt Lake City, UT, USA (2016)
[16] Terao, J.; Wadahama, A.; Matono, A.; Tada, T.; Watanabe, S.; Seki, S.; Fujihara, T.; Tsuji, Y., Design principle for increasing charge mobility of \(\pi \)-conjugated polymers using regularly localized molecular orbitals, Nature Comm., 4, 1691 (2013)
[17] Tada, T., Wave-packet multi-scale simulations based on a non-linear tight-binding Hamiltonian for carrier transport in \(\pi \)-conjugated polymers, Mater. Chem. Front., 2, 1351 (2018)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.