Volume 14, Number 4, Winter 2006
VARIABLE-STEP VARIABLE-ORDER 3-STAGE
SOLVER OF ORDER 4 TO 14
TRUONG NGUYEN-BA, HEMZA YAGOUB, YU ZHANG AND
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 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 of 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.