Numerical methods for ordinary differential equations

Algorithms approximate solutions of ordinary differential equations by advancing computed values across discrete steps. They trade computational cost against accuracy, stability, and the ability to capture important behavior.

Euler method: A one-step method that advances an approximate solution using the derivative at the beginning of each step. Its simple update rule exposes the basic idea behind stepping through an initial-value problem.
Ordinary differential equation: An equation relating an unknown function of one independent variable to its derivatives. These algorithms target equations whose unknown functions depend on a single independent variable.
Stiff differential equation: A differential equation whose numerical solution can require unusually small steps for stability, despite slow changes in the solution. Stiffness often forces explicit methods to take impractically small steps.
Analytical solution of a differential equation: An exact expression or mathematical characterization of a differential equation's solution. Numerical methods approximate solutions where an exact representation may be unavailable or impractical.
Backward error analysis: The study of the nearby problem that a computed numerical result solves exactly. It distinguishes a small residual or perturbation from accuracy relative to the original model.
Runge–Kutta methods: A family of one-step methods that combine derivative evaluations within each step to achieve higher accuracy. Multiple slope samples improve accuracy without requiring higher derivatives.
Initial value problem: A differential equation together with values of the unknown function specified at one point. Most step-by-step methods advance from the supplied starting value.
Implicit method: A numerical method in which the new solution value is defined through an equation involving that unknown value. Implicit updates often remain stable at step sizes that destabilize explicit schemes.
Finite difference method: A method that approximates derivatives with differences on a discrete grid, commonly for differential equations. Finite differences are a separate discretization approach, especially for boundary-value and partial differential equations.
Global error: The accumulated difference between a numerical solution and the exact solution across a computation. Local accuracy alone does not reveal how far the final computed trajectory may drift.

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