A Variable-Step Double-Integration Multi-Step Integrator

A new method of numerical integration is presented here, the variable-step Störmer-Cowell method. The method uses error control to regulate the step size, so larger step sizes can be taken when possible, and is double-integration, so only one evaluation per step is necessary when integrating second-order differential equations. The method is not variable-order, because variable-order algorithms require a second evaluation. The variable-step Störmer-Cowell method is designed for space surveillance applications, which require numerical integration methods to track orbiting objects accurately. Because of the large number of objects being processed, methods that can integrate the equations of motion as fast as possible while maintaining accuracy requirements are desired. The force model used for earth-orbiting objects is quite complex and computa-tionally expensive, so methods that minimize the force model evaluations are needed. The new method is compared to the fixed-step Gauss-Jackson method, as well as a method of analytic step regulation (s-integration), and the variable-step variable-order Shampine-Gordon integrator. Speed and accuracy tests of these methods indicate that the new method is comparable in speed and accuracy to s-integration in most cases, though the variable-step Störmer-Cowell method has an advantage over s-integration when drag is a significant factor. The new method is faster than the Shampine-Gordon integrator, because the Shampine-Gordon integrator uses two evaluations per step, and is biased toward keeping the step size constant. Tests indicate that both the new variable-step Störmer-Cowell method and s-integration have an advantage over the fixed-step Gauss-Jackson method for orbits with eccentricities greater than 0.15. Acknowledgments This work could not have been completed without the help and support of many individuals. First, I would like to acknowledge my late advisor, Dr. Frederick Lutze. Dr. Lutze provided support and encouragement whenever it was needed, which was quite often. He is missed dearly by both his students and colleagues. Next, I must thank Liam Healy, who gave direction to this work and provided constant assistance. Some of this work was originally presented in conference papers which he co-authored. Liam has acted as a mentor to me over the years, and I owe much of what I've accomplished to his support. I would also like to thank Christopher Hall for taking over as my advisor and quickly coming up to speed on my work. The other members of my committee, Lee Johnson, Hanspeter Schaub and Craig Woolsey, have also provided assistance through lengthy discussion of this work. for providing the …