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Hermite-Birkhoff-Obrechkoff four-stage four-step ODE solver of order 14 with quantized step size. (English) Zbl 1153.65071

Summary: 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 \(y{^{\prime}}=f(t,y)\) with initial conditions \(y(t_{0})=y_{0}\). Its formula uses \(y{^{\prime}}, y{^{\prime\prime}}\) and \(y^{\prime \prime \prime }\) 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 ordinary differential equation 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/\(^{\sim}\)remi.

MSC:

65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
68W30 Symbolic computation and algebraic computation
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
65L70 Error bounds for numerical methods for ordinary differential equations

Software:

ode113; Matlab; HBOQ(14)4
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References:

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