RK4 Derivation#

We integrate the ODE system \(\dot{\mathbf{x}} = \mathbf{v}\), \(\dot{\mathbf{v}} = \mathbf{a}(\mathbf{x})\) with fixed time step \(\Delta t\).

Classical RK4 stages for a generic \(\dot{\mathbf{y}} = \mathbf{f}(\mathbf{y})\) are: \( \begin{aligned} \mathbf{k}_1 &= \mathbf{f}(\mathbf{y}_n),\\ \mathbf{k}_2 &= \mathbf{f}\!\left(\mathbf{y}_n + \tfrac{\Delta t}{2}\,\mathbf{k}_1\right),\\ \mathbf{k}_3 &= \mathbf{f}\!\left(\mathbf{y}_n + \tfrac{\Delta t}{2}\,\mathbf{k}_2\right),\\ \mathbf{k}_4 &= \mathbf{f}\!\left(\mathbf{y}_n + \Delta t\,\mathbf{k}_3\right),\\ \mathbf{y}_{n+1} &= \mathbf{y}_n + \tfrac{\Delta t}{6}\left(\mathbf{k}_1 + 2\mathbf{k}_2 + 2\mathbf{k}_3 + \mathbf{k}_4\right). \end{aligned} \)

In ElectroSim, we apply RK4 to \((\mathbf{x},\mathbf{v})\) with acceleration recomputed from temporary states. Only non-fixed particles are written back.

Error:

  • Local truncation error \(\mathcal{O}(\Delta t^5)\), global error \(\mathcal{O}(\Delta t^4)\).

  • For conservative systems, RK4 does not conserve energy exactly; small drifts can appear.

See implementation: electrosim.simulation.physics.rk4_integrate().

References: :cite:butcher_ode.

This Page