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Runge–Kutta Methods
In numerical analysis, the Runge–Kutta methods ( ) are a family of Explicit and implicit methods, implicit and explicit iterative methods, List of Runge–Kutta methods, which include the Euler method, used in temporal discretization for the approximate solutions of simultaneous nonlinear equations. These methods were developed around 1900 by the German mathematicians Carl Runge and Wilhelm Kutta. The Runge–Kutta method The most widely known member of the Runge–Kutta family is generally referred to as "RK4", the "classic Runge–Kutta method" or simply as "the Runge–Kutta method". Let an initial value problem be specified as follows: : \frac = f(t, y), \quad y(t_0) = y_0. Here y is an unknown function (scalar or vector) of time t, which we would like to approximate; we are told that \frac, the rate at which y changes, is a function of t and of y itself. At the initial time t_0 the corresponding y value is y_0. The function f and the initial conditions t_0, y_0 are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Runge–Kutta Methods
Runge–Kutta methods are methods for the numerical solution of the ordinary differential equation :\frac = f(t, y). Explicit and implicit methods, Explicit Runge–Kutta methods take the form :\begin y_ &= y_n + h \sum_^s b_i k_i \\ k_1 &= f(t_n, y_n), \\ k_2 &= f(t_n+c_2h, y_n+h(a_k_1)), \\ k_3 &= f(t_n+c_3h, y_n+h(a_k_1+a_k_2)), \\ &\;\;\vdots \\ k_i &= f\left(t_n + c_i h, y_n + h \sum_^ a_ k_j\right). \end Stages for Explicit and implicit methods, implicit methods of s stages take the more general form, with the Explicit and implicit methods#Computation, solution to be found over all s :k_i = f\left(t_n + c_i h, y_n + h \sum_^ a_ k_j\right). Each method listed on this page is defined by its Butcher tableau, which puts the coefficients of the method in a table as follows: : \begin c_1 & a_ & a_& \dots & a_\\ c_2 & a_ & a_& \dots & a_\\ \vdots & \vdots & \vdots& \ddots& \vdots\\ c_s & a_ & a_& \dots & a_ \\ \hline & b_1 & b_2 & \dots & b_s\\ \end For ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
