Runge–Kutta methods

[2] These methods were developed around 1900 by the German mathematicians Carl Runge and Wilhelm Kutta.

, so that the differential equation is equivalent to a simple integral, then RK4 is Simpson's rule.

It is given by where[6] To specify a particular method, one needs to provide the integer s (the number of stages), and the coefficients aij (for 1 ≤ j < i ≤ s), bi (for i = 1, 2, ..., s) and ci (for i = 2, 3, ..., s).

[7] These data are usually arranged in a mnemonic device, known as a Butcher tableau (after John C. Butcher): A Taylor series expansion shows that the Runge–Kutta method is consistent if and only if There are also accompanying requirements if one requires the method to have a certain order p, meaning that the local truncation error is O(hp+1).

[12] In general, however, it remains an open problem what the precise minimum number of stages

Some values which are known are:[13] The provable bound above then imply that we can not find methods of orders

The work of Butcher also proves that 7th and 8th order methods have a minimum of 9 and 11 stages, respectively.

[11][12] An example of an explicit method of order 6 with 7 stages can be found in Ref.

Its tableau is[18] A slight variation of "the" Runge–Kutta method is also due to Kutta in 1901 and is called the 3/8-rule.

[19] The primary advantage this method has is that almost all of the error coefficients are smaller than in the popular method, but it requires slightly more FLOPs (floating-point operations) per time step.

The method proceeds as follows: The numerical solutions correspond to the underlined values.

Adaptive methods are designed to produce an estimate of the local truncation error of a single Runge–Kutta step.

Thanks to this, estimating the error has little or negligible computational cost compared to a step with the higher-order method.

This results in an (almost), optimal step size, which saves computation time.

Moreover, the user does not have to spend time on finding an appropriate step size.

The Butcher tableau for this kind of method is extended to give the values of

These two schemes also have the symplectic-preserving properties when the original equation is derived from a conservative classical mechanical system, i.e. when

Explicit Runge–Kutta methods are generally unsuitable for the solution of stiff equations because their region of absolute stability is small; in particular, it is bounded.

[25] This issue is especially important in the solution of partial differential equations.

The consequence of this difference is that at every step, a system of algebraic equations has to be solved.

This can be contrasted with implicit linear multistep methods (the other big family of methods for ODEs): an implicit s-step linear multistep method needs to solve a system of algebraic equations with only m components, so the size of the system does not increase as the number of steps increases.

Its Butcher tableau is: The trapezoidal rule is a collocation method (as discussed in that article).

[31] It follows from the formula that r is the quotient of two polynomials of degree s if the method has s stages.

Explicit methods have a strictly lower triangular matrix A, which implies that det(I − zA) = 1 and that the stability function is a polynomial.

[32] The numerical solution to the linear test equation decays to zero if | r(z) | < 1 with z = hλ.

Thus, it is of interest to study quotients of polynomials of given degrees that approximate the exponential function the best.

In contrast, the order of A-stable linear multistep methods cannot exceed two.

Dahlquist (1963) proposed the investigation of stability of numerical schemes when applied to nonlinear systems that satisfy a monotonicity condition.

We develop the derivation[38] for the Runge–Kutta fourth-order method using the general formula with

If we define: and for the previous relations we can show that the following equalities hold up to