Integrable algorithm
Integrable algorithms are the name for numerical algorithms that has 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
- ^ Nakamura, Y. (2004, March). A new approach to numerical algorithms in terms of integrable systems. In International Conference on Informatics Research for Development of Knowledge Society Infrastructure, 2004. ICKS 2004. (pp. 194-205). IEEE.
- ^ N. J. Zabusky and M. D. Kruskal, Phys. Rev. Lett. 15 (1965) 240-243.
- ^ Sogo, K. (1993). Toda molecule equation and quotient-difference method. Journal of the Physical Society of Japan, 62(4), 1081-1084.
- ^ Iwasaki, M., & Nakamura, Y. (2006). Accurate computation of singular values in terms of shifted integrable schemes. Japan journal of industrial and applied mathematics, 23(3), 239.
- ^ Papageorgiou, Grammaticos and Ramani (1993). Integrable Lattices and Convergence Acceleration Algorithms, Phys. Lett. A 179, 111-115.
- ^ Chang, X. K., He, Y., Hu, X. B., & Li, S. H. (2018). A new integrable convergence acceleration algorithm for computing Brezinski-Durbin-Redivo-Zaglia’s sequence transformation via Pfaffians. Numerical Algorithms, 1-20.
- ^ Nakamura, Y. (2001). Algorithms associated with arithmetic, geometric and harmonic means and integrable systems. Journal of Computational and Applied Mathematics, 131(1-2), 161-174.
- ^ Chu, M. T. (2008). Linear algebra algorithms as dynamical systems. Acta Numerica, 17, 1-86.
- ^ R. Hirota, J. Phys. Soc. Jpn. 43 (1977) 4116-4124.
- ^ R. Hirota, J. Phys. Soc. Jpn. 43 (1977) 2074-2078.
- ^ R. Hirota, J. Phys. Soc. Jpn. 43 (1977) 2079-2086.
- ^ R. Hirota, J. Phys. Soc. Jpn. 45 (1978) 321-332.
- ^ R. Hirota, J. Phys. Soc. Jpn. 46 (1979) 312-319.
- ^ M. J. Ablowitz and J. F. Ladik, J. Math. Phys. 16 (1975) 598-603.
- ^ M. J. Ablowitz and J. F. Ladik, J. Math. Phys. 17 (1976) 1011-1018.
- ^ M. J. Ablowitz and J. F. Ladik, Stud. Appl. Math. 55 (1977) 213-229.
- ^ M. J. Ablowitz and J. F. Ladik, Stud. Appl. Math. 57 (1977) 1-12.
- ^ M. J. Ablowitz and H. Segur, Solitons and Inverse Scattering Transform, (Society for Industrial and Applied Mathematics, Philadelphia, 1981).
- ^ T. R. Taha and M. J. Ablowitz, J. Comput. Phys., 55 (1984), 192-202.
- ^ T. R. Taha and M. J. Ablowitz, J. Comput. Phys., 55 (1984), 203-230.
- ^ T. R. Taha and M. J. Ablowitz, J. Comput. Phys., 55 (1984), 231-253.
- ^ T. R. Taha and M. J. Ablowitz, J. Comput. Phys., 55 (1988), 540-548.