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Higher-order splitting algorithms for solving the nonlinear Schrödinger equation and their instabilities.
1Department of Physics, Texas A&M University, College Station, Texas 77843, USA.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|February 1, 2008
Summary
Splitting algorithms for the nonlinear Schrödinger equation can be unstable due to noise growth. This numerical instability is unavoidable for continuous wave functions but can be managed for discrete ones.
Area of Science:
- Computational physics
- Quantum mechanics
Background:
- The real-time nonlinear Schrödinger equation is crucial in quantum mechanics.
- Splitting algorithms offer exact solutions for kinetic and potential energy terms.
Purpose of the Study:
- To analyze the numerical instability of splitting algorithms for the nonlinear Schrödinger equation.
- To investigate the behavior of noise and its impact on stability.
Main Methods:
- Numerical verification of fourth-order convergence for splitting algorithms.
- Detailed error analysis of noise propagation and spectral properties.
- Investigation of instability conditions for continuum and discrete wave functions.
Main Results:
- Splitting algorithms exhibit latent numerical instability despite energy conservation.
- Instability arises from the exponential growth of high-wave-number noise, following the Bogoliubov spectrum.
- This instability is unavoidable for continuum wave functions.
Conclusions:
- Numerical instability in splitting algorithms for the nonlinear Schrödinger equation is linked to noise amplification.
- For discrete wave functions, stability can be maintained by adhering to the condition Deltatkmax2 <= 2pi.
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