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:

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