Variable-step variable-order 3-stage Hermite-Birkhoff-Obrechkoff DDE solver of order 4 to 14

Variable-step variable-order 3-stage Hermite–Birkhoff–Obrechkoff methods of order 4 to 14, denoted by HBO(4-14)3, are constructed for solving non-stiff systems of first-order differential equations of the form y′ = f(x, y), y(x0) = y0. These methods use 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-type and Runge–Kutta-type order conditions which are reorganized into linear Vandermonde-type systems. Fast and stable algorithms are developed for solving these systems to obtain Hermite–Birkhoff interpolation polynomials in terms of generalized Lagrange basis functions. The order and stepsize of these methods are controlled by four local error estimators. When programmed in Matlab, HBO(4-14)3 is superior to Matlab’s ode113 in solving several problems often used to test higher order ODE solvers on the basis the number of steps, CPU time, and maximum global error. It is also superior to the variable step 3-stage HBO(14)3 of order 14 on some problems. Programmed in C++, HBO(4-14)3 is superior to DP(8,7)13M in solving expensive equations over a long period of time. Mathematical Subject Classification. Primary 65L06; Secondary 65D05, 65D30