---
tag: 558XADG64M
kind: concept
name: Numerical methods for ordinary differential equations
msc: [65L05, 65L06, 65L20]
sources:
  - url: https://en.wikipedia.org/wiki/Numerical_methods_for_ordinary_differential_equations
same_as:
  wikidata: Q541182
  wikipedia: Numerical methods for ordinary differential equations
---

## Summary

Numerical methods for ordinary differential equations compute approximate values of the solution of an [[initial value problem|0FXDW992DS]] $y' = f(t, y)$, $y(t_0) = y_0$, at a finite set of times.
Most [[ordinary differential equations|0NT64JV96A]] have no solution in closed form, so these methods are the main way to solve them in science and engineering.
The two big families are one-step methods, such as the [[Euler method|0JARTXX4XP]] and the [[Runge–Kutta methods|7DBPJKYC6C]], and [[linear multistep methods|BX4D85A79P]].
Their theory studies consistency, measured by the [[local truncation error|T220HF0562]], together with stability and convergence.
It pays special attention to [[stiff equations|RR5FCG92PN]] and to [[adaptive step size|6MREWBZR7R]] control.

## Explanations

### L1 human

A differential equation tells how fast a quantity changes, given its current value.
A numerical method starts from the known initial value and takes small time steps, using the rate of change to predict the next value each time.
For $y' = y$, $y(0) = 1$, steps of length $0.1$ with the rule "new value equals old value times $1.1$" give $1.1^{10} \approx 2.594$ at time $1$, close to the true value $e \approx 2.718$.
Better methods sample the rate of change more cleverly and get much closer with the same number of steps.