Variable-step variable-order 2-stage Hermite-Birkhoff-Obrechkoff (HBO) methods, HBO(p)2, of order p = 3 to 14, named HBO(3-14)2, are constructed for solving nonstiff first-order differential equations. Forcing an expansion of the numerical solution to agree with a Taylor expansion of the true solution leads to multistep and Runge-Kutta type order conditions which are reorganized into linear Vandermonde-type systems of HBO type. Fast algorithms are developed for solving these systems in O( p2) operations to obtain Hermite-Birkhoff interpolation polynomials in terms of generalized Lagrange basis functions. The order and step size of these methods are controlled by four local error estimators. For numerical computation the lower order 3 is raised to 4 since HBO(4-14)2 produces better results. When programmed in Matlab, HBO (4-14) 2 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. On the other hand, HBO (4-14)2 and the Dormand-Prince 13-stage nested Runge-Kutta pair DP(8,7)13M are programmed in C. In this case, DP(8,7) uses less CPU time, have smaller maximum global error but require a larger number of function evaluations than HBO(4-14)2 on nonexpensive problems. However, for expensive equations, such as the Cubicwave, HBO(4-14)2 is superior. Compared with previous results obtained by the 3-stage HBO(4-14)3 on Van der Pol equations with increasing value of epsilon, the new HBO(4-14)2 finally dominates.
| Identifer | oai:union.ndltd.org:uottawa.ca/oai:ruor.uottawa.ca:10393/27746 |
| Date | January 2008 |
| Creators | Zhuang, Yuchuan |
| Publisher | University of Ottawa (Canada) |
| Source Sets | Université d’Ottawa |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | 56 p. |
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