ODE Methods Euler & Runge-Kutta Reference Cheat Sheet

Key Facts

• An initial value problem has the form $y'=f(t,y)$ with $y(t_0)=y_0$, and a numerical method estimates values $y_n \approx y(t_n)$.
• The time grid is defined by $t_n=t_0+nh$, where $h$ is the step size and $n$ is a nonnegative integer.
• Euler's method updates by $y_{n+1}=y_n+h f(t_n,y_n)$, using the slope at the beginning of the interval.
• Improved Euler's method uses $k_1=f(t_n,y_n)$, $k_2=f(t_n+h,y_n+h k_1)$, and $y_{n+1}=y_n+\frac{h}{2}(k_1+k_2)$.
• The midpoint RK2 method uses $k_1=f(t_n,y_n)$, $k_2=f(t_n+\frac{h}{2},y_n+\frac{h}{2}k_1)$, and $y_{n+1}=y_n+h k_2$.
• Classical RK4 uses $k_1=f(t_n,y_n)$, $k_2=f(t_n+\frac{h}{2},y_n+\frac{h}{2}k_1)$, $k_3=f(t_n+\frac{h}{2},y_n+\frac{h}{2}k_2)$, $k_4=f(t_n+h,y_n+h k_3)$, and $y_{n+1}=y_n+\frac{h}{6}(k_1+2k_2+2k_3+k_4)$.
• Euler's method has global error $O(h)$, RK2 methods have global error $O(h^2)$, and classical RK4 has global error $O(h^4)$ under standard smoothness assumptions.
• Halving $h$ approximately halves Euler global error, divides RK2 global error by $4$, and divides RK4 global error by $16$ when the asymptotic error model applies.

Vocabulary

Initial value problem

A differential equation with a starting condition, usually written as $y'=f(t,y)$ and $y(t_0)=y_0$.

Step size

The fixed or variable spacing $h=t_{n+1}-t_n$ between consecutive numerical time values.

Euler's method

A first-order one-step method that estimates the next value using $y_{n+1}=y_n+h f(t_n,y_n)$.

Runge-Kutta method

A one-step method that combines several slope estimates from $f(t,y)$ to produce a more accurate update.

Local truncation error

The error made in one step when the method starts from the exact value, often described as a power of $h$.

Global error

The accumulated difference between the numerical value $y_n$ and the exact value $y(t_n)$ after many steps.