A model of the pretzel knot is described. Explicit Runge-Kutta methods have been studied for over a century and have applications in the sciences as well as mathematical software such as Matlab's ode45 solver. We have taken a new look at fourth- and fifth-order Runge-Kutta methods by utilizing techniques based on Gröbner bases to design explicit fourth-order Runge-Kutta formulas with step doubling and a family of (4,5) formula pairs that minimize the higher-order truncation error. Gröbner bases, useful tools for eliminating variables, also helped to reveal patterns among the error terms. A Matlab program based on step doubling was then developed to compare the accuracy and efficiency of fourth-order Runge-Kutta formulas with that of ode45.

Author Bio

Stephen Dupal is a senior at Rose-Hulman Institute of Technology, majoring in computer engineering and minoring in mathematics. He plans to obtain a Master's of Electrical and Computer Engineering while at Rose-Hulman. He and Michael Yoshizawa completed this paper as part of a Mathematics Research Experience for Undergraduates (REU) at Iowa State University under the supervision of Dr. Roger Alexander. Stephen presented the work at a talk during the Ohio State University Young Mathematicians Conference this past summer. The REU was sponsored by NSF grant DMS-0353880.

Michael Yoshizawa is a senior at Pomona College, located in Claremont, CA. He is majoring in mathematics and plans to attend graduate school in mathematics as well. The work for this paper was completed with Stephen Dupal during a Mathematics Research Experience for Undergraduates at Iowa State University in the summer of 2006, sponsored by NSF grant DMS-0353880.