OOF2: The Manual


4th order Runge-Kutta (RK4) — Fourth order Runge-Kutta time stepping.





RK4 is a TimeStepper that implements the classic fourth order Runge-Kutta method for solving ordinary differential equations. The error on each step is of order \(\Delta t^5\).

Given a vector \(\phi_n\) of unknowns (i.e. Field values in OOF2) at time \(t_n\), and the first order differential equation

      \[ \frac{d\phi}{dt} = f(\phi, t) \]

the fourth order Runge-Kutta estimate for \(\phi_{n+1}\) is given by

      k_1 &= \Delta t f(\phi_n, t_n) \\
      k_2 &= \Delta t f(\phi_n+\frac{k_1}{2}, t_n+\frac{\Delta t}{2}) \\
      k_3 &= \Delta t f(\phi_n+\frac{k_2}{2}, t_n+\frac{\Delta t}{2}) \\
      k_4 &= \Delta t f(\phi_n+k_3, t_n+\Delta t) \\
      \phi_{n+1} &= \phi_n + \frac{k_1}{6} + \frac{k_2}{3} + \frac{k_3}{3}
      + \frac{k_4}{6}

where \(\Delta t =

RK4 can be applied to second order equations by using equation (6.141).

Because the method is explicit (\(\phi_{n+1}\) doesn't appear as an argument to \(f\)), equation (6.158) doesn't require a nonlinear solver even if \(f\) is nonlinear.