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Heun's method is a second-order method with two stages. It is also known as the explicit trapezoid rule, improved Euler's method, or modified Euler's method: It is also known as the explicit trapezoid rule, improved Euler's method, or modified Euler's method:
The midpoint method computes + so that the red chord is approximately parallel to the tangent line at the midpoint (the green line). In numerical analysis , a branch of applied mathematics , the midpoint method is a one-step method for numerically solving the differential equation ,
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.
In turn, an explicit method can be obtained from this if, for example, the unknown + on the right-hand side of the equation is approximated using the explicit Euler method, the so-called Heun method. [4] All these methods and all other generalizations have the basic idea of one-step methods in common: the step
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 .
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]
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.
Midpoint method — a second-order method with two stages; Heun's method — either a second-order method with two stages, or a third-order method with three stages; Bogacki–Shampine method — a third-order method with four stages (FSAL) and an embedded fourth-order method