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Integrable algorithms are numerical algorithms that rely on basic ideas from the mathematical theory of integrable systems.[1]

Background

The theory of integrable systems has advanced with the connection between numerical analysis. For example, the discovery of solitons came from the numerical experiments to the KdV equation by Norman Zabusky and Martin David Kruskal.[2] Today, various relations between numerical analysis and integrable systems have been found (Toda lattice and numerical linear algebra,[3][4] discrete soliton equations and series acceleration[5][6]), and studies to apply integrable systems to numerical computation are rapidly advancing.[7][8]

Integrable difference schemes

Generally, it is hard to accurately compute the solutions of nonlinear differential equations due to its non-linearity. In order to overcome this difficulty, R. Hirota has made discrete versions of integrable systems with the viewpoint of "Preserve mathematical structures of integrable systems in the discrete versions".[9][10][11][12][13]

At the same time, Mark J. Ablowitz and others have not only made discrete soliton equations with discrete Lax pair but also compared numerical results between integrable difference schemes and ordinary methods.[14][15][16][17][18] As a result of their experiments, they have found that the accuracy can be improved with integrable difference schemes at some cases.[19][20][21][22]

References

  1. ^ Nakamura, Y. (2004). A new approach to numerical algorithms in terms of integrable systems. International Conference on Informatics Research for Development of Knowledge Society Infrastructure. IEEE. pp. 194–205. doi:10.1109/icks.2004.1313425. ISBN 0-7695-2150-9.
  2. ^ Zabusky, N. J.; Kruskal, M. D. (1965-08-09). "Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States". Physical Review Letters. 15 (6). American Physical Society (APS): 240–243. Bibcode:1965PhRvL..15..240Z. doi:10.1103/physrevlett.15.240. ISSN 0031-9007.
  3. ^ Sogo, Kiyoshi (1993-04-15). "Toda Molecule Equation and Quotient-Difference Method". Journal of the Physical Society of Japan. 62 (4). Physical Society of Japan: 1081–1084. Bibcode:1993JPSJ...62.1081S. doi:10.1143/jpsj.62.1081. ISSN 0031-9015.
  4. ^ Iwasaki, Masashi; Nakamura, Yoshimasa (2006). "Accurate computation of singular values in terms of shifted integrable schemes". Japan Journal of Industrial and Applied Mathematics. 23 (3). Springer Science and Business Media LLC: 239–259. doi:10.1007/bf03167593. ISSN 0916-7005. S2CID 121824363.
  5. ^ Papageorgiou, V.; Grammaticos, B.; Ramani, A. (1993). "Integrable lattices and convergence acceleration algorithms". Physics Letters A. 179 (2). Elsevier BV: 111–115. Bibcode:1993PhLA..179..111P. doi:10.1016/0375-9601(93)90658-m. ISSN 0375-9601.
  6. ^ Chang, Xiang-Ke; He, Yi; Hu, Xing-Biao; Li, Shi-Hao (2017-07-01). "A new integrable convergence acceleration algorithm for computing Brezinski–Durbin–Redivo-Zaglia's sequence transformation via pfaffians". Numerical Algorithms. 78 (1). Springer Science and Business Media LLC: 87–106. doi:10.1007/s11075-017-0368-z. ISSN 1017-1398. S2CID 4974630.
  7. ^ Nakamura, Yoshimasa (2001). "Algorithms associated with arithmetic, geometric and harmonic means and integrable systems". Journal of Computational and Applied Mathematics. 131 (1–2). Elsevier BV: 161–174. Bibcode:2001JCoAM.131..161N. doi:10.1016/s0377-0427(00)00316-2. ISSN 0377-0427.
  8. ^ Chu, Moody T. (2008-04-25). "Linear algebra algorithms as dynamical systems". Acta Numerica. 17. Cambridge University Press (CUP): 1–86. doi:10.1017/s0962492906340019. ISSN 0962-4929. S2CID 8746366.
  9. ^ Hirota, Ryogo (1977-10-15). "Nonlinear Partial Difference Equations. I. A Difference Analogue of the Korteweg-de Vries Equation". Journal of the Physical Society of Japan. 43 (4). Physical Society of Japan: 1424–1433. Bibcode:1977JPSJ...43.1424H. doi:10.1143/jpsj.43.1424. ISSN 0031-9015.
  10. ^ Hirota, Ryogo (1977-12-15). "Nonlinear Partial Difference Equations. II. Discrete-Time Toda Equation". Journal of the Physical Society of Japan. 43 (6). Physical Society of Japan: 2074–2078. Bibcode:1977JPSJ...43.2074H. doi:10.1143/jpsj.43.2074. ISSN 0031-9015.
  11. ^ Hirota, Ryogo (1977-12-15). "Nonlinear Partial Difference Equations III; Discrete Sine-Gordon Equation". Journal of the Physical Society of Japan. 43 (6). Physical Society of Japan: 2079–2086. Bibcode:1977JPSJ...43.2079H. doi:10.1143/jpsj.43.2079. ISSN 0031-9015.
  12. ^ Hirota, Ryogo (1978-07-15). "Nonlinear Partial Difference Equations. IV. Bäcklund Transformation for the Discrete-Time Toda Equation". Journal of the Physical Society of Japan. 45 (1). Physical Society of Japan: 321–332. Bibcode:1978JPSJ...45..321H. doi:10.1143/jpsj.45.321. ISSN 0031-9015.
  13. ^ Hirota, Ryogo (1979-01-15). "Nonlinear Partial Difference Equations. V. Nonlinear Equations Reducible to Linear Equations". Journal of the Physical Society of Japan. 46 (1). Physical Society of Japan: 312–319. Bibcode:1979JPSJ...46..312H. doi:10.1143/jpsj.46.312. ISSN 0031-9015.
  14. ^ Ablowitz, M. J.; Ladik, J. F. (1975). "Nonlinear differential−difference equations". Journal of Mathematical Physics. 16 (3). AIP Publishing: 598–603. Bibcode:1975JMP....16..598A. doi:10.1063/1.522558. ISSN 0022-2488.
  15. ^ Ablowitz, M. J.; Ladik, J. F. (1976). "Nonlinear differential–difference equations and Fourier analysis". Journal of Mathematical Physics. 17 (6). AIP Publishing: 1011–1018. Bibcode:1976JMP....17.1011A. doi:10.1063/1.523009. ISSN 0022-2488.
  16. ^ Ablowitz, M. J.; Ladik, J. F. (1976). "A Nonlinear Difference Scheme and Inverse Scattering". Studies in Applied Mathematics. 55 (3). Wiley: 213–229. doi:10.1002/sapm1976553213. ISSN 0022-2526.
  17. ^ Ablowitz, M. J.; Ladik, J. F. (1977). "On the Solution of a Class of Nonlinear Partial Difference Equations". Studies in Applied Mathematics. 57 (1). Wiley: 1–12. doi:10.1002/sapm19775711. ISSN 0022-2526.
  18. ^ Ablowitz, Mark J.; Segur, Harvey (1981). Solitons and the Inverse Scattering Transform. Philadelphia: Society for Industrial and Applied Mathematics. doi:10.1137/1.9781611970883. ISBN 978-0-89871-174-5.
  19. ^ Taha, Thiab R; Ablowitz, Mark J (1984). "Analytical and numerical aspects of certain nonlinear evolution equations. I. Analytical". Journal of Computational Physics. 55 (2). Elsevier BV: 192–202. Bibcode:1984JCoPh..55..192T. doi:10.1016/0021-9991(84)90002-0. ISSN 0021-9991.
  20. ^ Taha, Thiab R; Ablowitz, Mark I (1984). "Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation". Journal of Computational Physics. 55 (2). Elsevier BV: 203–230. Bibcode:1984JCoPh..55..203T. doi:10.1016/0021-9991(84)90003-2. ISSN 0021-9991.
  21. ^ Taha, Thiab R; Ablowitz, Mark I (1984). "Analytical and numerical aspects of certain nonlinear evolution equations. III. Numerical, Korteweg-de Vries equation". Journal of Computational Physics. 55 (2). Elsevier BV: 231–253. Bibcode:1984JCoPh..55..231T. doi:10.1016/0021-9991(84)90004-4. ISSN 0021-9991.
  22. ^ Taha, Thiab R; Ablowitz, Mark J (1988). "Analytical and numerical aspects of certain nonlinear evolution equations IV. Numerical, modified Korteweg-de Vries equation". Journal of Computational Physics. 77 (2). Elsevier BV: 540–548. Bibcode:1988JCoPh..77..540T. doi:10.1016/0021-9991(88)90184-2. ISSN 0021-9991.

See also

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