Numerical methods for ordinary differential equations

Illustration of numerical integration for the differential equation $y'=y,y(0)=1.$

  Blue: Euler method

  Green: Midpoint method

  Red: Exact solution: ${\textstyle y=e^{t}}$.

The step size is $h=1.0$.

The same illustration for $h=0.25.$ The midpoint method converges faster than the Euler method, as $h\to 0$.

Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals.

Many differential equations cannot be solved exactly. For practical purposes, however, a numerical approximation to the solution is often sufficient. The algorithms studied here can be used to compute such an approximation. An alternative method is to use techniques from calculus to obtain a series expansion of the solution.

Ordinary differential equations occur in many scientific disciplines, including physics, chemistry, biology, and economics. In addition, some numerical methods for partial differential equations convert the partial differential equation into a system of ordinary differential equations, which can then be solved numerically.