Runge–Kutta methods

Runge–Kutta methods approximate solutions to ordinary differential equations by combining several evaluations of the derivative within each step. Their coefficients determine accuracy, stability, and computational cost.

Explicit Runge–Kutta method: A Runge–Kutta method whose stages depend only on earlier stages in the same step. Its sequential slope evaluations show the standard construction behind many practical formulas.
Ordinary differential equation: An equation relating an unknown function of one variable to its derivatives. Runge–Kutta methods target initial-value problems for this class of equations.
Euler method: A first-order numerical method that advances an ordinary differential equation using one derivative evaluation per step. It is the simplest slope-based baseline that Runge–Kutta methods improve.
Linear multistep method: A numerical method that advances a differential-equation solution using values from several previous time steps. Unlike Runge–Kutta schemes, it reuses past solution values rather than relying only on stages within a step.
Butcher tableau: A compact array of coefficients specifying the stages and weights of a Runge–Kutta method. It records exactly how a method combines its stage derivatives.
Initial value problem: A differential equation together with the value of its solution at a specified starting point. Each numerical step advances an approximation from the given initial state.
Classical fourth-order Runge–Kutta method: A four-stage explicit Runge–Kutta method with fourth-order accuracy for sufficiently smooth solutions. Its four weighted slope samples make the family concrete and widely recognizable.
Adams–Bashforth method: An explicit linear multistep method that combines derivative values from current and earlier steps. It can use fewer new derivative evaluations after startup than a comparable Runge–Kutta scheme.
Embedded Runge–Kutta method: A Runge–Kutta scheme that computes two approximations of different orders from shared stages. The difference between its paired estimates supplies a local error indicator.
Numerical integration: The use of numerical procedures to approximate mathematical quantities or solutions. Runge–Kutta methods belong to the broader toolkit for approximating differential-equation solutions.

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