Numerics

The numerical methods of example_pkg: the equations each one implements, its order of accuracy, assumptions and references. Each card is declared on the code that implements it, checked by the tests, and also shown on the method's API page.

Time steps

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

Grid

midpoint_integral

functionsource

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

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

2

Assumes

• f twice continuously differentiable on [a, b]

References

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