J. S. C. Prentice

J. S. C. Prentice

We study the effect of global error control in the numerical solution of Hamiltonian systems. In particular, we apply the RKQ algorithm in the numerical solution of a Hamiltonian system. This algorithm is designed to provide stepwise control of both local and global error. A test problem demonstrates the error control features of RKQ. Good results are obtained, despite the fact that explicit Runge-Kutta methods have been used in RKQ, rather than symplectic Runge-Kutta methods. This simply emphasizes the value of stepwise global error control, as per the RKQ algorithm.

J. S. C. Prentice

We study the relationship between local and global error in Runge-Kutta methods for initial-value problems in ordinary differential equations. We show that local error control by means of local extrapolation does not equate to global error control. Our analysis shows that the global error of the higher-order solution is propagated under iteration, and this can cause an uncontrolled increase in the global error of the lower-order solution. We find conditions under which global error control occurs during the initial stages of the RK integration, but even in such a case the global error is likely to eventually exceed the user-defined tolerance.

J. S. C. Prentice

We determine the pointwise error in Hermite interpolation by numerically solving an appropriate differential equation, derived from the error term itself. We use this knowledge to approximate the error term by means of a polynomial, which is then added to the original Hermite polynomial to form a more accurate approximation. An example demonstrates that improvements in accuracy are significant.