What is Numerical methods for ordinary differential equations?
'''Numerical methods for ordinary differential equations''' are computational schemes to obtain approximate solutions of [[ordinary differential equation]]s (ODEs). ==Background== Since ODEs appeared in science, many mathematicians have studied how to solve them.<ref>[[Vladimir Arnold|Arnolʹd, V. I.]], Ordinary differential equations. Springer.</ref><ref>Wolfgang Walter, Ordinary differential equations. Springer.</ref><ref>Logemann, H., & Ryan, E. P. (2014). Ordinary differential equations: Analysis, qualitative theory and control. Springer.</ref><ref>Chicone, C. (2006). Ordinary differential equations with applications. Springer Science & Business Media.</ref> 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,<ref>Butcher, J. C. (1996). A history of Runge-Kutta methods. Applied Numerical Mathematics, 20(3), 247-260.</ref> 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: {{div col|rules=yes}} <!--alphabetical order--> * Bulirsch-Stoer algorithm<ref>Monroe, J. L. (2002). Extrapolation and the Bulirsch-Stoer algorithm. Physical Review E, 65(6), 066116.</ref> * Euler's method (named after [[Leonhard Euler]]) and their variants ** Backward Euler method<ref>Peskin, C. S., & Schlick, T. (1989). Molecular dynamics by the Backward‐Euler method. Communications on pure and applied mathematics, 42(7), 1001-1031.</ref> ** Semi-implicit Euler method ** Euler-Maruyama method<ref>Emma Gau (2020). Euler–Maruyama Method (https://www.mathworks.com/matlabcentral/fileexchange/69430-euler-maruyama-method), MATLAB Central File Exchange. Retrieved May 24, 2020.</ref> * Exponential integrator<ref>Hochbruck, M., & Ostermann, A. (2010). Exponential integrators. Acta Numerica, 19, 209-286.</ref><ref>Al-Mohy, A. H., & Higham, N. J. (2011). Computing the action of the matrix exponential, with an application to exponential integr