example_pkg

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A tiny package, so the example site has an API reference to render.

Classes

Name	Description
Drift	Move the position at the current velocity.
Grid	A uniform grid of cells on [start, stop].
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.

Functions

Name	Description
linspace_midpoints	The midpoints of cells uniform cells on [start, stop].
midpoint_integral	The integral of f on [start, stop] by the composite midpoint rule.

Modules

• gridA uniform 1D grid.
• schemesTime steps for a particle in a force field: kick, drift and their leapfrog.

Driftclass#

Re-exported from example_pkg.schemesView source

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#

View source

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

Advance the position only.

Gridclassdataclass#

Re-exported from example_pkg.gridView source

class Grid(start: float, stop: float, cells: int)

A uniform grid of cells on [start, stop].

Attributes

Name	Type	Description
start	float	Left end of the domain.
stop	float	Right end of the domain.
cells	int	Number of cells.

cellsattributeinstance attribute#

View source

cells: int

startattributeinstance attribute#

View source

start: float

stopattributeinstance attribute#

View source

stop: float

spacingproperty#

View source

spacing: float

The cell width $h = (b - a) / n$.

nodesmethod#

View source

def nodes() -> np.ndarray

The cells + 1 cell boundaries.

Returns

np.ndarray

The node coordinates.

Kickclass#

Re-exported from example_pkg.schemesView source

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#

View source

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

Advance the velocity only.

Leapfrogclass#

Re-exported from example_pkg.schemesView source

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#

View source

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

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

stepmethod#

View source

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

Advance (x, v) by one leapfrog step.

linspace_midpointsfunction#

Re-exported from example_pkg.gridView source

def linspace_midpoints(start: float, stop: float, cells: int) -> np.ndarray

The midpoints of cells uniform cells on [start, stop].

Parameters

Name	Type	Description
start	float	Left end of the domain.
stop	float	Right end of the domain.
cells	int	Number of cells.

Returns

np.ndarray

An array of length cells.

midpoint_integralfunction#

Re-exported from example_pkg.gridView source

def midpoint_integral(f, start: float, stop: float, cells: int) -> float

The integral of f on [start, stop] by the composite midpoint rule.

Parameters

Name	Type	Description
f		The integrand, called on an array of points.
start	float	Left end of the domain.
stop	float	Right end of the domain.
cells	int	Number of cells.

Returns

float

The approximate integral.

Numerics

rule

$$\int_a^b f(x)\,dx \approx h \sum_{i=0}^{n-1} f\left(\frac{x_i + x_{i+1}}{2}\right)$$

error

$$\left|E\right| \le \frac{(b - a) h^2}{24} \max_{[a, b]} |f''|$$

Order of accuracy: 2.

Assumptions:

• f twice continuously differentiable on [a, b]

References:

• Süli and Mayers, An Introduction to Numerical Analysis (2003), section 7.2