Numerical methods for ordinary differential equations

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Numerical methods for ordinary differential equations are computational schemes to obtain approximate solutions of ordinary differential equations (ODEs).

Background[change | change source]

Since ODEs appeared in science, many mathematicians have studied how to solve them.[1][2][3][4] However, only few of them can be mathematically solved. This is why numerical methods are needed. One of the most famous methods are the Runge-Kutta methods,[5] but it doesn't work for some ODEs (especially nonlinear ODEs). This is why new ODE solvers are developed. The following list includes frequently used methods:

  • Bulirsch-Stoer algorithm[6]
  • Euler's method (named after Leonhard Euler) and their variants
    • Backward Euler method[7]
    • Semi-implicit Euler method
    • Euler-Maruyama method[8]
  • Exponential integrator[9][10]
  • Leapfrog method
  • Linear multistep methods
  • Shooting method
  • Symplectic integrator[11][12][13][14]
  • Taylor series method[15][16]

Validated Numerics for ODEs[change | change source]

Not only approximate solvers, but the study to "verify the existence of solution by computers" is also active. This study is needed because numerically obtained solutions could be phantom solutions (fake solutions). This kind of incident is already reported.[17][18] The popular methods are based on the shooting method or spectral methods.[19][20] Today, European research teams[21][22][23][24][25][26][27][28][29] and Japanese experts[30][31] are working on this topic.

ODEs and Related Topics Studied in the Context of Validated Numerics[change | change source]

Related Software[change | change source]

References[change | change source]

  1. Arnolʹd, V. I., Ordinary differential equations. Springer.
  2. Wolfgang Walter, Ordinary differential equations. Springer.
  3. Logemann, H., & Ryan, E. P. (2014). Ordinary differential equations: Analysis, qualitative theory and control. Springer.
  4. Chicone, C. (2006). Ordinary differential equations with applications. Springer Science & Business Media.
  5. Butcher, J. C. (1996). A history of Runge-Kutta methods. Applied Numerical Mathematics, 20(3), 247-260.
  6. Monroe, J. L. (2002). Extrapolation and the Bulirsch-Stoer algorithm. Physical Review E, 65(6), 066116.
  7. Peskin, C. S., & Schlick, T. (1989). Molecular dynamics by the Backward‐Euler method. Communications on pure and applied mathematics, 42(7), 1001-1031.
  8. Emma Gau (2020). Euler–Maruyama Method (, MATLAB Central File Exchange. Retrieved May 24, 2020.
  9. Hochbruck, M., & Ostermann, A. (2010). Exponential integrators. Acta Numerica, 19, 209-286.
  10. Al-Mohy, A. H., & Higham, N. J. (2011). Computing the action of the matrix exponential, with an application to exponential integrators. SIAM Journal on Scientific Computing, 33(2), 488-511.
  11. Hairer, E., Lubich, C., & Wanner, G. (2006). Geometric numerical integration: structure-preserving algorithms for ordinary differential equations. Springer Science & Business Media.
  12. Symplectic integrators: An introduction, American Journal of Physics 73, 938 (2005); Denis Donnelly.
  13. Y. B. Suris, Hamiltonian Runge-Kutta type methods and their variational formulation (1990) Matematicheskoe modelirovanie, 2(4), 78-87.
  14. Iserles, A., & Quispel, G. R. W. (2016). Why geometric integration?. arXiv preprint arXiv:1602.07755.
  15. Hirayama, H. (2002). Solution of ordinary differential equations by Taylor series method. JSIAM, 12, 1-8.
  16. Hirayama, H. (2015). Performance of a Higher-Order Numerical Method for Solving Ordinary Differential Equations by Taylor Series. In Integral Methods in Science and Engineering (pp. 321-328). Birkhäuser, Cham.
  17. Breuer, B., Plum, M., & McKenna, P. J. (2001). "Inclusions and existence proofs for solutions of a nonlinear boundary value problem by spectral numerical methods." In Topics in Numerical Analysis (pp. 61–77). Springer, Vienna.
  18. Gidas, B., Ni, W. M., & Nirenberg, L. (1979). "Symmetry and related properties via the maximum principle." Communications in Mathematical Physics, 68(3), 209–243.
  19. Lloyd N. Trefethen (2000) Spectral Methods in MATLAB. SIAM, Philadelphia, PA.
  20. D. Gottlieb and S. Orzag (1977) "Numerical Analysis of Spectral Methods : Theory and Applications", SIAM, Philadelphia, PA.
  21. 21.0 21.1 21.2 Lohner,R.J.,Enclosing the Solution of Ordinary lnitial and Boundary Value Problems, Computer arithmetic:Scientific Computation and Programming Languages,Kaucher,E.,Kulisch,U., Ullrich,Ch.(eds.), B.G.Teubner,Stuttgart (1987), 255−286.
  22. Rihm, R. (1994). Interval methods for initial value problems in ODEs. Topics in Validated Computations, 173-207.
  23. Hungria, A., Lessard, J. P., & Mireles-James, J. D. (2014). Radii polynomial approach for analytic solutions of differential equations: Theory, examples, and comparisons. Math. Comp.
  24. Nedialkov, N. S., Jackson, K. R., & Pryce, J. D. (2001). An effective high-order interval method for validating existence and uniqueness of the solution of an IVP for an ODE. Reliable Computing, 7(6), 449-465.
  25. Corliss, G. F. (1989). Survey of interval algorithms for ordinary differential equations. Applied Mathematics and Computation, 31, 112-120.
  26. Nedialkov, N. S. (2000). Computing rigorous bounds on the solution of an initial value problem for an ordinary differential equation (Ph.D. thesis). University of Toronto.
  27. Eijgenraam, P. (1981). The solution of initial value problems using interval arithmetic: formulation and analysis of an algorithm. MC Tracts.
  28. Nedialkov, N. S., & Jackson, K. R. (1999). An interval Hermite-Obreschkoff method for computing rigorous bounds on the solution of an initial value problem for an ordinary differential equation. Reliable Computing, 5(3), 289-310.
  29. Nedialkov, N. S., Jackson, K. R., & Corliss, G. F. (1999). Validated solutions of initial value problems for ordinary differential equations. Applied Mathematics and Computation, 105(1), 21-68.
  30. Berz, M., & Makino, K. (1998). Verified integration of ODEs and flows using differential algebraic methods on high-order Taylor models. Reliable computing, 4(4), 361-369.
  31. Kashiwagi, M. (1995). Numerical Validation for Ordinary Differential Equations using Power Series Arithmetic. In Numerical Analysis Of Ordinary Differential Equations And Its Applications (pp. 213-218).
  32. Takayasu, A., Matsue, K., Sasaki, T., Tanaka, K., Mizuguchi, M., & Oishi, S. I. (2017). Numerical validation of blow-up solutions of ordinary differential equations. Journal of Computational and Applied Mathematics, 314, 10-29.
  33. Matsue, K., & Takayasu, A. (2019). Rigorous numerics of blow-up solutions for ODEs with exponential nonlinearity. arXiv preprint arXiv:1902.01842.
  34. Hassard, B., Zhang, J., Hastings, S. P., & Troy, W. C. (1994). A computer proof that the Lorenz equations have “chaotic” solutions. Applied Mathematics Letters, 7(1), 79-83.
  35. Mischaikow, K., & Mrozek, M. (1995). Chaos in the Lorenz equations: a computer-assisted proof. en:Bulletin of the American Mathematical Society, 32(1), 66-72.
  36. Mischaikow, K., & Mrozek, M. (1998). Chaos in the Lorenz equations: A computer assisted proof. Part II: Details. en:Mathematics of Computation, 67(223), 1023-1046.
  37. Mischaikow, K., Mrozek, M., & Szymczak, A. (2001). Chaos in the lorenz equations: A computer assisted proof part iii: Classical parameter values. Journal of Differential Equations, 169(1), 17-56.
  38. Galias, Z., & Zgliczyński, P. (1998). Computer assisted proof of chaos in the Lorenz equations. Physica D: Nonlinear Phenomena, 115(3-4), 165-188.
  39. Tucker, W. (1999). The Lorenz attractor exists. Comptes Rendus de l'Académie des Sciences-Series I-Mathematics, 328(12), 1197-1202.
  40. Zgliczynski, P. (1997). Computer assisted proof of chaos in the Rössler equations and in the Hénon map. Nonlinearity, 10(1), 243.
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  49. Wouwer, A. V., Saucez, P., & Vilas, C. (2014). Simulation of Ode/Pde Models with MATLAB®, OCTAVE and SCILAB: Scientific and Engineering Applications. Springer.
  50. Houcque, D. (2008). Applications of MATLAB: Ordinary differential equations (ODE). Robert R. McCormick School of Engineering and Applied Science-Northwestern University, Evanston.
  51. Shampine, L. F., & Reichelt, M. W. (1997). The matlab ode suite. SIAM Journal on Scientific Computing, 18(1), 1-22.
  52. Ashino, R., Nagase, M., & Vaillancourt, R. (2000). Behind and beyond the MATLAB ODE suite. Computers & Mathematics with Applications, 40(4-5), 491-512.
  53. Gladwell, I. (1979). Initial value routines in the NAG library. ACM Transactions on Mathematical Software (TOMS), 5(4), 386-400.
  54. Gladwell, I. (1979). The development of the boundary-value codes in the ordinary differential equations chapter of the NAG library. In Codes for Boundary-Value Problems in Ordinary Differential Equations (pp. 122-143). Springer, Berlin, Heidelberg.
  55. Berzins, M., Brankin, R. W., & Gladwell, I. (1988). The stiff integrators in the NAG library. ACM SIGNUM Newsletter, 23(2), 16-23.
  56. Baumann, G. (2013). Symmetry analysis of differential equations with Mathematica®. Springer Science & Business Media.
  57. Abell, M. L., & Braselton, J. P. (2016). Differential equations with Mathematica. Academic Press.
  58. Gray, A., Mezzino, M., & Pinsky, M. A. (1997). Introduction to ordinary differential equations with Mathematica: an integrated multimedia approach. Springer.
  59. Ross, C. C. (2013). Differential equations: an introduction with Mathematica®. Springer Science & Business Media.

Further reading[change | change source]

  • Mitsui, T., & Shinohara, Y. (1995). Numerical analysis of ordinary differential equations and its applications. World Scientific.
  • Iserles, A. (2009). A first course in the numerical analysis of differential equations. Cambridge University Press.
  • Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems, Berlin, New York: Springer-Verlag.
  • Wanner, G. & Hairer, E. (1996), Solving ordinary differential equations II: Stiff and differential-algebraic problems (2nd ed.). Springer Berlin Heidelberg.
  • Butcher, John C. (2008), Numerical Methods for Ordinary Differential Equations, New York: John Wiley & Sons.
  • John D. Lambert, Numerical Methods for Ordinary Differential Systems, John Wiley & Sons, Chichester, 1991.
  • Deuflhard, P., & Bornemann, F. (2012). Scientific computing with ordinary differential equations. Springer Science & Business Media.
  • Shampine, L. F. (2018). Numerical solution of ordinary differential equations. Routledge.
  • Dormand, John R. (1996), Numerical Methods for Differential Equations: A Computational Approach, Boca Raton: CRC Press.

Other websites[change | change source]