Time steps

The equations of example_pkg.schemes, declared on the code that implements them and checked by the tests.

Drift

classsource

Move the position at the current velocity.

update

$$x^{n+1} = x^n + v^n\,\Delta t$$

Assumes

• velocity constant over the step

Kick

classsource

Change the velocity by the acceleration at the current position.

update

$$v^{n+1} = v^n + a(x^n)\,\Delta t$$

Leapfrog

classsource

Half a kick, a drift and half a kick: the velocity Verlet scheme.

half kick

$$v^{n+1/2} = v^n + \tfrac{1}{2} a(x^n)\,\Delta t$$

drift

$$x^{n+1} = x^n + v^{n+1/2}\,\Delta t$$

kick

$$v^{n+1} = v^{n+1/2} + \tfrac{1}{2} a(x^{n+1})\,\Delta t$$

energy

$$\left|H(x^n, v^n) - H(x^0, v^0)\right| = \mathcal{O}(\Delta t^2) \quad \text{for all } n$$

Order

x: 2, v: 2

Assumes

• the acceleration depends on the position only

References

• Hairer, Lubich and Wanner, Geometric Numerical Integration (2006), section I.1.4

Built from

• Kick
• Drift