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.
Dupal, Stephen and Yoshizawa, Michael
"Design and Optimization of Explicit Runge-Kutta Formulas,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 8:
1, Article 8.
Available at: https://scholar.rose-hulman.edu/rhumj/vol8/iss1/8