Runge-Kutta order versus stages

The textbook version of the Runge-Kutta method for solving differential equations has 4 stages and has 4th order error. For lower order versions of RK the number of stages s also matches the order of the error p. But in order to achieve error on the order of p ≥ 5, you need more than p stages. This is known as the Butcher barrier. Before going any further, let’s back up and say what we mean by stages and by order. Stages The number of stages in an RK method to solve the equation is the number of evaluations of the function f on the right-hand side. For example, the textbook RK4 method estimates the solution at each step by where which requires four stages, i.e. four evaluations of f. Order A differential equation solver is said to have order p if the local error, the error after one step of size h, is O(hp + 1). Then after solving an ODE over a period of time T with N = T/h steps, the global error is O(hp). So, for example, if p = 4, you would expect that cutting your step size h in…