Variable-step variable-order 3-stage Hermite-Birkhoff-Obrechkoff ODE solver of order 4 to 14 with a C program

Abstract

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 nonstiff 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's method. 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. Fast algorithms are developed for solving these systems to obtain Hermite-Birkhoff interpolation polynomials in terms of generalized Lagrange basis functions. The new methods have larger regions of absolute stability than Adams-Bashforth-Moulton methods of comparable orders in PECE mode. The order and stepsize of these methods are controlled by four local error estimators. When programmed in Matlab, HBO(4-14)3 are superior to Matlab's ode113 in solving several problems often used to test higher order ODE solvers on the basis of the number of function evaluations, 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. When programmed in C, HBO(4-14)3 is superior to the Dormand-Prince Runge-Kutta nested pair DP(8,7)13M in solving expensive equations over a long period of time. HBO(4-14)3 has been implemented in C. The C program for HBO(4-14)3 is an important contribution of this thesis.