example_pkg.schemes

Time steps for a particle in a force field: kick, drift and their leapfrog.

Classes

Name	Description
Drift	Move the position at the current velocity.
Kick	Change the velocity by the acceleration at the current position.
Leapfrog	Half a kick, a drift and half a kick: the velocity Verlet scheme.
Scheme	A one-step scheme for x'' = a(x).

Driftclass#

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class Drift(Scheme)

Bases: Scheme

Move the position at the current velocity.

Numerics

update

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

Assumptions:

• velocity constant over the step

stepmethod#

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def step(x, v, dt: float, acceleration: Callable) -> tuple

Advance the position only.

Kickclass#

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class Kick(Scheme)

Bases: Scheme

Change the velocity by the acceleration at the current position.

Numerics

update

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

stepmethod#

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def step(x, v, dt: float, acceleration: Callable) -> tuple

Advance the velocity only.

Leapfrogclass#

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class Leapfrog(Scheme)

Bases: Scheme

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

Numerics

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 of accuracy: x: 2, v: 2.

Assumptions:

• the acceleration depends on the position only

References:

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

Built from:

• example_pkg.schemes.Kick
• example_pkg.schemes.Drift

runmethod#

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def run(x, v, dt: float, steps: int, acceleration: Callable) -> np.ndarray

Take steps steps and return the positions, (steps + 1,).

stepmethod#

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def step(x, v, dt: float, acceleration: Callable) -> tuple

Advance (x, v) by one leapfrog step.

Schemeclass#

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class Scheme(NumericalMethod)

Bases: NumericalMethod

A one-step scheme for x'' = a(x).

stepmethod#

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def step(x, v, dt: float, acceleration: Callable) -> tuple

Advance (x, v) by dt.