Nine-stage multi-derivative Runge-Kutta method of order 12. (English) Zbl 1265.65134

Nguyen-Ba, Truong; Božić, Vladan; Kengne, Emmanuel; Vaillancourt, Rémi

Publ. Inst. Math., Nouv. Sér. 86(100), 75-96 (2009).

Author’s abstract: A nine-stage multi-derivative Runge-Kutta method of order 12, called HBT(12)9, is constructed for solving nonstiff systems of first-order differential equations (ODEs) of the form \(y'=f(x,y)\), \(y(x_0)=y_0\). The method uses \(y'\) and higher derivatives \(y^{(2)}\) to \(y^{(6)}\) as in Taylor methods and is combined with a \(9\)-stage Runge-Kutta method. Forcing an expansion of the numerical solution to agree with a Taylor expansion of the true solution leads to order conditions which are reorganized into Vandermonde-type linear systems whose solutions are the coefficients of the method. The stepsize is controlled by means of the derivatives \(y^{(3)}\) to \(y^{(6)}\). The new method has a larger interval of absolute stability than Dormand-Prince’s DP(8,7)13M and is superior to DP(8,7)13M and the Taylor method of order 12 in solving several problems often used to test high-order ODE solvers on the basis of the number of steps, CPU time, maximum global error of position and energy. Numerical results show the benefits of adding high-order derivatives to Runge-Kutta methods.

Reviewer: Boško Jovanović (Beograd)

MSC:

65L06	Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
34A34	Nonlinear ordinary differential equations and systems
65L05	Numerical methods for initial value problems involving ordinary differential equations
65L50	Mesh generation, refinement, and adaptive methods for ordinary differential equations
65L70	Error bounds for numerical methods for ordinary differential equations
65L20	Stability and convergence of numerical methods for ordinary differential equations

Keywords:

general linear method for non-stiff ODE; Hermite-Birkhoff method; Taylor method; maximum global error; number of function evaluations; CPU time; stepsize control; stability; numerical results