Time Stepping Methods

Time stepping methods are methods for calculating how a value (or dimensioned values) evolves over time. Such an evolution should be expressed as a system of equations of the form:

The implemented methods are:

• Explicit Euler (first order) (Euler)
• Crank-Nicolson (second order) (CrankNicolson)
• Second order Runge Kutta (RK2)
• Third order Runge Kutta (RK3)
• Fourth order Runge Kutta (RK4)

These classes all contain an update method which carries out one time step of the algorithm.

Explicit Euler method

• Scheme: \(x^{n+1} = x^{n} + dt f(x^{n+1})\)

• Convergence order : 1.

Crank-Nicolson method

• Scheme: \(x^{k} = x^{n} + \frac{dt}{2} \left( f(x^{n}) + f(x^k) \right)\) and \(x^{n+1} = x^{k+1}\) once converged.

• Convergence order : 2.

RK2 method

• Scheme: \(x^{n+1} = x^{n} + dt k_2\)

• with

  • \(k_1 = f(x^{n})\),
  • \(k_2 = f(x^{n} + \frac{dt}{2} k_1)\),

• Convergence order : 3.

RK3 method

• Scheme: \(x^n = x^{n+1} - \frac{dt}{6} \left( k_1 + 4 k_2 + k_3 \right)\)

• with

  • \(k_1 = f(x^{n})\),
  • \(k_2 = f(x^{n} + \frac{dt}{2} k_1)\),
  • \(k_3 = f(x^{n} + dt( 2k_2 - k_1))\).

• Convergence order : 3.

RK4 method

• Scheme: \(x^{n+1} = x^{n} + \frac{dt}{6} \left( k_1 + 2 k_2 + 2 k_3 + k_4\right)\)

• with

  • \(k_1 = f(x^{n+1})\),
  • \(k_2 = f(x^{n+1} + \frac{dt}{2} k_1)\),
  • \(k_3 = f(x^{n+1} + \frac{dt}{2} k_2)\),
  • \(k_4 = f(x^{n+1} + dt k_3)\).

• Convergence order : 4.