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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.

MSC:

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

Citations:

Zbl 1342.65232

Software:

mb4-nls2d; GitHub
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References:

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