Numerical methods for ordinary differential equations

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

Background

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

See also: Validated numerics

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

• Blow-up solutions^[32][33]
• Lorentz equation^{[34][35][36][37][38][39]}
• Rossler equation^[40]

Related Software

• Chebfun^{[41][42][43][44]}
• COSY INFINITY Archived 2020-09-24 at the Wayback Machine^[45][46]
• INTLAB^[21][47] and kv^[21][48] are interval arithmetic libraries which include ODE solvers
• MATLAB^{[49][50][51][52]} - made by MathWorks
• NAG library^[53][54][55]
• Wolfram Mathematica^{[56][57][58][59]} - made by Wolfram Research