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The backward differentiation formula (BDF) is a family of implicit methods for the numerical integration of ordinary differential equations.They are linear multistep methods that, for a given function and time, approximate the derivative of that function using information from already computed time points, thereby increasing the accuracy of the approximation.
This differs from the (forward) Euler method in that the forward method uses (,) in place of (+, +). The backward Euler method is an implicit method: the new approximation y k + 1 {\displaystyle y_{k+1}} appears on both sides of the equation, and thus the method needs to solve an algebraic equation for the unknown y k + 1 {\displaystyle y_{k+1}} .
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.
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.
Linear multistep methods are used for the numerical solution of ordinary differential equations.Conceptually, a numerical method starts from an initial point and then takes a short step forward in time to find the next solution point.
The Runge–Kutta–Fehlberg method has two methods of orders 5 and 4; it is sometimes dubbed RKF45 . Its extended Butcher Tableau is: / / / / / / / / / / / / / / / / / / / / / / / / / / The first row of b coefficients gives the fifth-order accurate solution, and the second row has order four.
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The stability function of an explicit Runge–Kutta method is a polynomial, so explicit Runge–Kutta methods can never be A-stable. [32] If the method has order p, then the stability function satisfies () = + (+) as . Thus, it is of interest to study quotients of polynomials of given degrees that approximate the exponential function the best.