Convergence of a three-dimensional quantum lattice Boltzmann scheme towards solutions of the Dirac equation
Abstract
We investigate the convergence properties of a three-dimensional quantum lattice Boltzmann scheme for the Dirac equation. These schemes were constructed as discretizations of the Dirac equation based on operator splitting to separate the streaming along the three coordinate axes, but their output has previously only been compared against solutions of the Schrödinger equation. The Schrödinger equation arises as the non-relativistic limit of the Dirac equation, describing solutions that vary slowly compared with the Compton frequency. We demonstrate first-order convergence towards solutions of the Dirac equation obtained by an independent numerical method based on fast Fourier transforms and matrix exponentiation.
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Consistency of the (1+1)-dimensional Dirac QW has been observed numerically in [9] and in (3+1)-dimensions in [10-12].
“It has been observed numerically in + (1 1) -dimensions in [9] and in + (3 1)-dimensions in [10-12].”
From The Dirac equation as a quantum walk: higher dimensions, observational convergence (2014) · cites this paper as [11]
Cites
Lattice Boltzmann equation for quantum mechanics
Succi, S., & Benzi, R. (1993) · Physica D: Nonlinear Phenomena · doi:10.1016/0167-2789(93)90096-j · Cited by 217 · see citing works
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The Dirac equation as a quantum walk: higher dimensions, observational convergence
Arrighi, P., Nesme, V., & Forets, M. (2014) · Journal of Physics A: Mathematical and Theoretical · doi:10.1088/1751-8113/47/46/465302 · Cited by 72 · see citing works
- Authors
- Lapitski, D., & Dellar, P. J.
- Affiliations
- University of Oxford (GB)
- ORCID
- Dellar, Paul J.
- Venue
- Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
- Publisher
- The Royal Society
- Published
- 2011-06-13
- Volume
- 369
- Issue
- 1944
- Pages
- 2155-2163
- ISSN
- 1364-503X (ISSN-L)
- 1471-2962
- OpenAlex
- W2102961571
- Cited by
- 23
- References
- 1
- Citations by year
- 2024: 1 · 2023: 1 · 2022: 1 · 2019: 4 · 2018: 2 · 2017: 3 · 2015: 2 · 2014: 1 · 2013: 3 · 2012: 3
- FWCI
- 2.90
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences (publishedVersion)
- PubMed (publishedVersion)
- Topics
- Physical Sciences › Engineering › Computational Mechanics › Lattice Boltzmann Simulation Studies
- Physical Sciences › Mathematics › Applied Mathematics › Gas Dynamics and Kinetic Theory
- Physical Sciences › Mathematics › Numerical Analysis › Numerical methods for differential equations
- Keywords
- Dirac equation, convergence analysis, operator splitting, relativistic quantum mechanics, non-relativistic limit, Schrödinger equation, fast Fourier transform, matrix exponentiation
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