DormandPrince

Dormand–Prince refers to a family of explicit Runge–Kutta methods used for solving initial value problems in ordinary differential equations. These methods are designed to deliver high accuracy with an embedded error estimation that enables automatic adaptive control of the step size. The most widely used member is the Dormand–Prince 5(4) pair, often called DOPRI5, which provides a fifth-order accurate solution along with a fourth-order embedded estimate.

The method works by evaluating a sequence of slopes (stages) to compute both a high-order solution and a lower-order error estimate from the same set of stage evaluations. The coefficients form an explicit Runge–Kutta tableau, and the embedded estimator is used to judge step acceptance and to adjust the next step size to meet a prescribed error tolerance.

DOPRI5 has become a standard in numerical software due to its reliable error control and efficiency for non-stiff problems. It is implemented in many library routines and widely known for its balance of accuracy and computational cost. The Dormand–Prince family also includes higher-order embedded pairs, such as DOP853, which yields an eighth-order solution with a seventh-order embedded estimate, suitable for applications requiring greater precision at the expense of more function evaluations.

History and impact: The Dormand–Prince family was introduced by Gerald Dormand and Philip Prince in a foundational paper on explicit Runge–Kutta formulae. Since then, these methods have become a common default for general-purpose ODE solving in scientific and engineering computing, especially in environments that employ adaptive step size control to manage local truncation error.