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In mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value.
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.
In Itô calculus, the Euler–Maruyama method (also simply called the Euler method) is a method for the approximate numerical solution of a stochastic differential equation (SDE). It is an extension of the Euler method for ordinary differential equations to stochastic differential equations named after Leonhard Euler and Gisiro Maruyama. The ...
Forward-Backward Euler method The result of applying both the Forward Euler method and the Forward-Backward Euler method for a = 5 {\displaystyle a=5} and n = 30 {\displaystyle n=30} . In order to apply the IMEX-scheme, consider a slightly different differential equation:
Another simple explicit method of order 2, the improved Euler method, is obtained by the following consideration: A "mean" slope in the method step would be the slope of the solution 𝑦 in the middle of the step, i.e. at the point +. However, as the solution is unknown, it is approximated by an explicit Euler step with half the step size.
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. Multistep methods attempt ...
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 .