Runge-Kutta Integrators

Given a differential equation: \(\dot{x} = f\left(x, t\right)\) We can solve this with varying accuracy.

Notes:

• The derivation of Runge-Kutta uses Taylor series
• Software like Matlab uses an adaptive Runge-Kutta, known as RK2,3 or RK4,5. This will adapt the timestep within its parameters to reduce errors in high dynamics, or reduce computation time in low dynamics.

Forward Euler

\[x_{k+1} = x_k + \Delta t f\left(x_k, t_k\right)\]

• Local error is \(O(\Delta t^2)\)
• Global error is \(O(\Delta t)\) - i.e. It increases linearly.

Runge-Kutta 2nd Order (RK2)

Conceptually this finds the gradient halfway along the next euler step - and uses this to take a full step. \(\begin{align} x_{k+1} &= x_k + \Delta t f_2 \\ \\ f_1 &= f\left(x_k, t_k\right) \\ f_2 &= f\left(x_k + \frac{\Delta t}{2} f_1, t_k + \frac{\Delta t}{2}\right) \end{align}\)

• Local error is \(O(\Delta t^3)\)
• Global error is \(O(\Delta t^2)\)

Runge-Kutta 4th Order (RK4)

This expands the Taylor series further, taking additional gradients into account.

\[\begin{align} x_{k+1} &= x_k + \frac{\Delta t}{6}\left(f_1 + 2 f_2 + 2 f_3 + f_4\right) \\ \\ f_1 &= f\left(x_k, t_k\right) \\ f_2 &= f\left(x_k + \frac{\Delta t}{2}f_1, t_k + \frac{\Delta t}{2}\right) \\ f_3 &= f\left(x_k + \frac{\Delta t}{2}f_2, t_k + \frac{\Delta t}{2}\right) \\ f_4 &= f\left(x_k + \Delta t f_3, t_k + \Delta t\right) \end{align}\]

• Local error is \(O(\Delta t^5)\)
• Global error is \(O(\Delta t^4)\)