Dormand-Prince method

Another method that uses fourth- and fifth-order embedded pairs is the Dormand-Prince method. The Dormand-Prince method is more accurate than the Runge-Kutta Fehlberg method and it is used by the MATLAB ode45 solver. Both methods have in common that the difference between the fourth- and fifth-order accurate solutions is calculated to determine the error, and to adapt the step size. The error estimate, e +, for the step is... 

Consequently, the error estimate in the Dormand-Prince method is given by... 

The differential equation has been resolved with an algorithm which makes use of an explicit Runge-Kutta(4,5) formula, the so-called Dormand-Prince pair. The least square minimization was implemented by a large-scale algorithm based on the interior-reflective Newton method. All the computations have been achieved by MATLAB software. 

Example 5.4.3 The 5 order Dormand and Prince method can be extended to an half-explicit method. For this method s = 6 and the coefficients aij,bi, i,j — 1,... 6 are given in (4-2.11). The additional coefficients can be selected in various ways. Arnold ([Am98]) sets them 50, that the residual of the hidden constraint ( -constraint) is damped in an optimal way. This leads to the coefficients of the code... 

In this flow chart is the end-point of integration, is the start-point of integration and NSTEP is the number of steps. The computation of qh zu i = 1(1)3 is based on the Runge-Kutta-Nystrom method of Dormand and Prince 8(7) (see 9-10). 

The behavior of the Runge-Kutta-Nystrom Symplectic method of algebraic order four developed by Sanz-Serna and Calvo12 and the behavior of the classical partitioned multistep method is similar. These methods are much more efficient that the embedded Runge-Kutta method of Dormand and Prince 5(4) (see 13). 

The authors produced a phase-fitted RKN method. The produced method was based on the Dormand, El-Mikkawy and Prince RKN method of algebraic order four with four (three effective) stages. The efficiency of the new proposed method is shown via numerical results. 

Runge-Kutta-Nystrom method of Prince and Dormand. With these starting values we evaluate at Xi of the asymptotic region the phase shift 5i... 

Since the problem is treated as an initial-value problem, one needs y>i, i = 0(1)4 before starting a six-step method. From the initial condition, = 0. The values i = 1,2,. . . are computed using the high order Runge-Kutta-Nystrom method of Prince and Dormand. With these starting values we evaluate at x of the asymptotic region the phase shift di from the relation (51). 

The fifth algebraic order Runge-Kutta method of Dormand and Prince. 

---