# Runge-Kutta methods and Butcher tableau

If you know one numerical method for solving ordinary differential equations, it’s probably Euler’s method.

If you know two methods, the second is probably 4th order Runge-Kutta.

It’s standard in classes on differential equations or numerical analysis to present Euler’s method as conceptually simple but inefficient introduction, then to present Runge-Kutta as a complicated but efficient alternative.

Runge-Kutta methods are a huge family of numerical methods with a wide variety of trade-offs: efficiency, accuracy, stability, etc.

Euler’s method is a member of the Runge-Kutta family as are countless other variations.

You could devote a career to studying Runge-Kutta methods, and some people have.

Beneath the complexity and variety, all Runge-Kutta methods have a common form that can be summarized by a matrix and two vectors.

For explicit Runge-Kutta methods (ERK) the matrix is triangular, and for implicit Runge-Kutta methods (IRK) the matrix is full.

This summary of an RK method is known as a Butcher tableau, named after J.

C.

Butcher who classified RK methods.

“The” Runge-Kutta method For example, let’s start with what students often take to be “the” Runge-Kutta method.

This method approximates solutions to a differential equation of the form by where.