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In mathematics and computational science, Heun's method may refer to the improved [1] or modified Euler's method (that is, the explicit trapezoidal rule [2]), or a similar two-stage Runge–Kutta method. It is named after Karl Heun and is a numerical procedure for solving ordinary differential equations (ODEs) with a given initial value.
The explicit midpoint method is sometimes also known as the modified Euler method, [1] the implicit method is the most simple collocation method, and, applied to Hamiltonian dynamics, a symplectic integrator. Note that the modified Euler method can refer to Heun's method, [2] for further clarity see List of Runge–Kutta methods.
A simple predictor–corrector method (known as Heun's method) can be constructed from the Euler method (an explicit method) and the trapezoidal rule (an implicit method). Consider the differential equation ′ = (,), =, and denote the step size by .
For this reason, the Euler method is said to be a first-order method, while the midpoint method is second order. We can extrapolate from the above table that the step size needed to get an answer that is correct to three decimal places is approximately 0.00001, meaning that we need 400,000 steps.
Download QR code; Print/export Download as PDF; ... It is also known as the explicit trapezoid rule, improved Euler's method, or modified Euler's method:
This is the Euler method (or forward Euler method, in contrast with the backward Euler method, to be described below). The method is named after Leonhard Euler who described it in 1768. The Euler method is an example of an explicit method. This means that the new value y n+1 is defined in terms of things that are already known, like y n.
Single-step methods (such as Euler's method) refer to only one previous point and its derivative to determine the current value. Methods such as Runge–Kutta take some intermediate steps (for example, a half-step) to obtain a higher order method, but then discard all previous information before taking a second step.
In numerical analysis, the Runge–Kutta methods (English: / ˈ r ʊ ŋ ə ˈ k ʊ t ɑː / ⓘ RUUNG-ə-KUUT-tah [1]) are a family of implicit and explicit iterative methods, which include the Euler method, used in temporal discretization for the approximate solutions of simultaneous nonlinear equations. [2]