Numerical methods for ordinary differential equations

methods used to find numerical solutions of ordinary differential equations

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

BackgroundEdit

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 ODEsEdit

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 NumericsEdit

Related SoftwareEdit

ReferencesEdit

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  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 (https://www.mathworks.com/matlabcentral/fileexchange/69430-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); https://doi.org/10.1119/1.2034523 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.
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  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.
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  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.
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  56. Baumann, G. (2013). Symmetry analysis of differential equations with Mathematica®. Springer Science & Business Media.
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Further readingEdit

  • 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 websitesEdit