Higher-order splitting algorithms for solving the nonlinear Schroedinger equation and their instabilities

Journal Article · Thu Nov 15 04:00:00 EST 2007 · Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

DOI:https://doi.org/10.1103/PHYSREVE.76.056708· OSTI ID:21076276

Chin, Siu A ^[1]

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1. Department of Physics, Texas A and M University, College Station, Texas 77843 (United States)

Since the kinetic and potential energy terms of the real-time nonlinear Schroedinger equation can each be solved exactly, the entire equation can be solved to any order via splitting algorithms. We verified the fourth-order convergence of some well-known algorithms by solving the Gross-Pitaevskii equation numerically. All such splitting algorithms suffer from a latent numerical instability even when the total energy is very well conserved. A detail error analysis reveals that the noise, or elementary excitations of the nonlinear Schroedinger equation, obeys the Bogoliubov spectrum and the instability is due to the exponential growth of high-wave-number noises caused by the splitting process. For a continuum wave function, this instability is unavoidable no matter how small the time step. For a discrete wave function, the instability can be avoided only for {delta}tk{sub max}{sup 2} < or approx. 2{pi}, where k{sub max}={pi}/{delta}x.

• OSTI ID: 21076276
• Journal Information: Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, Journal Name: Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics Journal Issue: 5 Vol. 76; ISSN PLEEE8; ISSN 1063-651X
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ALGORITHMS
CONVERGENCE
ERRORS
EXCITATION
INSTABILITY
MATHEMATICAL SOLUTIONS
NOISE
NONLINEAR PROBLEMS
POTENTIAL ENERGY
SCHROEDINGER EQUATION
WAVE FUNCTIONS