Hermite–Birkhoff–Obrechkoff four-stage four-step ODE solver of order 14 with quantized step size

• Published in: Journal of computational and applied mathematics 2008-12, Vol.222 (2), p.608-621
• Main Authors: Nguyen-Ba, Truong, Sharp, Philip W., Vaillancourt, Rémi
• Format: Article
• Language: English
• Subjects: Algebra; Algebraic geometry; Comparing ODE solvers; CPU time; DP(8,7)13M; Exact sciences and technology; General linear method for non-stiff ODE’s; Global analysis, analysis on manifolds; Hermite–Birkhoff method; Mathematical analysis; Mathematics; Maximum global error; Number of function evaluations; Numerical analysis; Numerical analysis. Scientific computation; Obrechkoff method; Ordinary differential equations; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Vandermonde-type systems
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2007.12.003

A four-stage Hermite–Birkhoff–Obrechkoff method of order 14 with four quantized variable steps, denoted by HBOQ(14)4, is constructed for solving non-stiff systems of first-order differential equations of the form y ′ = f ( t , y ) with initial conditions y ( t 0 ) = y 0 . Its formula uses y ′ , y ″ and y ‴ as in Obrechkoff methods. Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads to multistep- and Runge–Kutta-type order conditions which are reorganized into linear Vandermonde-type systems. To reduce overhead, simple formulae are derived only once to obtain the values of Hermite–Birkhoff interpolation polynomials in terms of Lagrange basis functions for 16 quantized step size ratios. The step size is controlled by a local error estimator. When programmed in C ++, HBOQ(14)4 is superior to the Dormand–Prince Runge–Kutta pair DP(8,7)13M of order 8 in solving several problems often used to test higher order ODE solvers at stringent tolerances. When programmed in Matlab, it is superior to ode113 in solving costly problems, on the basis of the number of steps, CPU time, and maximum global error. The code is available on the URL www.site.uottawa.ca/~remi.