Time integration methods:
This example shows the motion of a particle with an initial velocity. Moreover, a constant gravitational acceleration is acting on the particle. In this special case an analytic solution can be computed as:
$$\begin{align*}
\mathbf x(t + \Delta t) &= \mathbf x(t) + \Delta t \mathbf v(t) + \frac12 \Delta t^2 \mathbf a^\text{grav} \\
\mathbf v(t + \Delta t) &= \mathbf v(t) + \Delta t \mathbf a^\text{grav} .
\end{align*}$$
The motion determined using this analytic solution is plotted in green and can be compared with different numerical solutions that are introduced in the following.
Explicit Euler
The explicit Euler method is a first-order accurate method for solving ordinary differential equations (ODEs):
$$\begin{align*}
\mathbf x(t + \Delta t) &= \mathbf x(t) + \Delta t \mathbf v(t) \\
\mathbf v(t + \Delta t) &= \mathbf v(t) + \Delta t \mathbf a(t).
\end{align*}$$
Symplectic Euler
The symplectic Euler method is also known as semi-implicit Euler since first the new velocity is determined and then the new position is computed using the new velocity value:
$$\begin{align*}
\mathbf v(t + \Delta t) &= \mathbf v(t) + \Delta t \mathbf a(t) \\
\mathbf x(t + \Delta t) &= \mathbf x(t) + \Delta t \mathbf v(t + \Delta t).
\end{align*}$$
The method is also first-order accurate but yields better results than the explicit Euler since it is a symplectic method.
Runge-Kutta 2
Typically numerical integration methods are defined for a differential equation determined by a function $\mathbf f(t, \mathbf s(t))$, where $\mathbf s$ defines the state of the system at time $t$. In our example the state is defined by the current position and velocity of the particle and the function is defined as:
$$\begin{equation*}
\mathbf f(t, \mathbf s(t)) = \begin{pmatrix} \dot{\mathbf x} \\ \dot{\mathbf v} \end{pmatrix}, \quad\quad \mathbf s(t) = \begin{pmatrix} \mathbf x(t) \\ \mathbf v(t) \end{pmatrix}
\end{equation*}$$
Using this function the second-order Runge-Kutta integration is defined as
$$\begin{align*}
\mathbf k_1 &= \Delta t \mathbf f(t, \mathbf s(t)) \\
\mathbf k_2 &= \Delta t \mathbf f(t + \frac12 \Delta t, \mathbf s(t) + \frac12 \mathbf k_1) \\
\mathbf s(t + \Delta t) &= \mathbf s(t) + \mathbf k_2.
\end{align*}$$
Note that the Runge-Kutta method has multiple stages (in our case 2) to achieve a higher-order accuracy. However, this approach is more expensive than the Euler methods since the function has to be evaluated once per stage.
|