Integration Methods¶

The following integration methods are included in ode:
• Euler’s method

• Backward Euler method

• Verlet method

The integration methods operate on systems of either first or second order differential equations. By convention $$X$$ is the vector containing the state variables of the system, $$f(t,X)$$ is a function returning either the first or second derivative of the system, and $$h$$ is the timestep.

The current state and derivative of the system are represented as lists.

Euler’s method¶

Euler’s method is an explicit method for solving a system of first order differential equations.

$f(t,X) = \dot{X}$
$X_{n+1} = X_n + h \cdot f(t_n, X_n)$

Backward Euler method¶

Euler’s method is an implicit method for solving a system of first order differential equations.

$f(t,X) = \dot{X}$
$X_{n+1} = X_n + h \cdot f(t_{n+1}, X_{n+1})$

Verlet method¶

The Verlet method, also called Störmer–Verlet method, is an explicit method for solving a system of second order differential equations. An initial velocity vector $$V_0$$ is needed as well as the initial condition $$X_0$$.

$f(t,X) = \ddot{X}$

The first step is calculated with:

$X_1 = X_0 + V_0 h + \frac{1}{2} f(X_0) h^2.$

Subsequent steps are calculated with:

$X_{n+1} = 2 X_{n} - X_{n-1} + f(t, X_n) h^2.$

If subsequent velocities are needed, they can be calculated with:

$V_n = \frac{X_{n+1} - X_{n-1}}{2 h}.$