Reducing the Uncertainty When Approximating the Solution of ODEs
One can reduce the uncertainty in the quality of an approximate solution of an ordinary differential equation (ODE) by implementing methods which have a more rigorous error control strategy and which deliver an approximate solution that is much more likely to satisfy the expectations of the user. We have developed such a class of ODE methods as well as a collection of software tools that will deliver a piecewise polynomial as the approximate solution and facilitate the investigation of various aspects of the problem that are often of as much interest as the approximate solution itself. We will introduce measures that can be used to quantify the reliability of an approximate solution and discuss how one can implement methods that, at some extra cost, can produce very reliable approximate solutions and therefore significantly reduce the uncertainty in the computed results.
KeywordsNumerical methods initial value problems ODEs reliable methods defect control
- 7.Gladwell, I., Shampine, L., Baca, L., Brankin, R.: Practical aspects of interpolation in Runge-Kutta codes. SIAM Journal of Scientific and Statistical Computing (8), 322–341 (1987)Google Scholar
- 10.Shampine, L.: Interpolation for Runge-Kutta methods. SIAM Journal of Numererical Analysis (22), 1014–1027 (1985)Google Scholar
- 11.Shampine, L.: Solving ODEs and DDEs with residual control. Applied Numererical Mathematics (52), 113–127 (2005)Google Scholar