Heun's method: Difference between revisions

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In [[mathematics]] and [[computational science]], '''Heun's method''' (alsomay calledrefer to the '''improved''' or '''modified Euler's method''' or(that is, the '''explicit trapezoidal rule'''<ref>

{{Citation | last1=Ascher | first1=Uri M. | last2=Petzold | first2=Linda R. | title=Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations | publisher=[[Society for Industrial and Applied Mathematics]] | ___location=Philadelphia | isbn=978-0-89871-412-8 | year=1998}}.</ref>), or a similar two-stage [[Runge–Kutta method]] method. It is named after [[Karl L. W. M. Heun]] and is a [[numerical analysis|numerical]] procedure for solving [[Ordinary_differential_equation|ordinary differential equations]] (ODEs) with a given [[Initial_value_problem|initial value]]. Both variants can be seen as extensions of the [[Euler method]] into two-stage second-order Runge–Kutta methods.

named after [[Karl L. W. M. Heun]], is a [[numerical analysis|numerical]] procedure for solving [[Ordinary_differential_equation|ordinary differential equations]] (ODEs) with a given [[Initial_value_problem|initial value]]. It can be seen as an extension of the [[Euler method]] into a two-stage second-order [[Runge–Kutta method]].

The procedure for calculating the numerical solution to the initial value problem via the improved Euler's method is:

:<math>y'(t) = f(t,y(t)), \qquad \qquad y(t_0)=y_0, </math>

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==Runge–Kutta method==

HeunThe improved Euler's method is a two-stage [[Runge–Kutta method]], and can be written using the [[Butcher tableau]] (after [[John C. Butcher]]):

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The other method referred to as Heun's method has the Butcher table<ref>

{{Citation | last1=Leader | first1=Jeffery J.| title=Numerical Analysis and Scientific Computation | publisher=[[Addison-Wesley]] | ___location=Boston | isbn=0-201-73499-0 | year=2004}}.</ref>:

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