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The backward Euler method is an implicit method, meaning that the formula for the backward Euler method has + on both sides, so when applying the backward Euler method we have to solve an equation. This makes the implementation more costly.
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]
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.
2 Illustration using the forward and backward Euler methods. 3 ... is a small time step), then, for an explicit method ... is rewritten as the sum of two ...
Two steps of the explicit Euler method. The simplest and most basic one-step method is the explicit Euler method, which was introduced by the Swiss mathematician and physicist Leonhard Euler in 1768 in his textbook Institutiones Calculi Integralis. [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.
Let us now apply Euler's method again with a different step size to generate a second approximation to y(t n+1). We get a second solution, which we label with a (). Take the new step size to be one half of the original step size, and apply two steps of Euler's method. This second solution is presumably more accurate.
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 .