||Exploring Runge-Kutta formulas with a computer algebra system
Runge-Kutta formulas are some of the workhorses of numerical methods for solving differential equations. However, they are extremely difficult to generate; the algebra involved can be very complicated indeed, and so their derivation is not included in undergraduate numerical texts. It is now standard, following with work of Butcher in the 1960’s and 70’s, to use the combinatorial theory of trees to simplify the algebra. More recently, however, several authors have shown that it is quite feasible to use a computer algebra system to generate Runge-Kutta formulas. This article shows that, using a computer-based approach, the formulas can be generated with very elementary means, using only the tools of elementary calculus and an open-source computer system to handle the messy algebra. This approach brings a formerly difficult operation into the realm of undergraduate mathematics.