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.