List of ODE solvers

Euler method

Algorithm 1 (Euler method)

Inputs A function \(f\) and an initial condition \(y_0\)

Output A sequence of approximations \(y_n\) to the solution \(y(t)\)

1. Set \(y_0 = y(0)\)

2. For \(n = 0, 1, 2, \ldots\) do

   1. Set \(y_{n+1} = y_n + hf(t_n, y_n)\)

Runge–Kutta method

Algorithm 2 (Runge–Kutta method)

Inputs A function \(f\) and an initial condition \(y_0\)

Output A sequence of approximations \(y_n\) to the solution \(y(t)\)

1. Set \(y_0 = y(0)\)

2. For \(n = 0, 1, 2, \ldots\) do

   1. Set \(k_1 = hf(t_n, y_n)\)

   2. Set \(k_2 = hf(t_n + \frac{h}{2}, y_n + \frac{k_1}{2})\)

   3. Set \(k_3 = hf(t_n + \frac{h}{2}, y_n + \frac{k_2}{2})\)

   4. Set \(k_4 = hf(t_n + h, y_n + k_3)\)

   5. Set \(y_{n+1} = y_n + \frac{1}{6}(k_1 + 2k_2 + 2k_3 + k_4)\)