Introduction to Numerical methods for ODEs#

Let’s assume that we want to find an analytical expression of a function that describes the displacement an object in time, denoted as $u(t)$. For simplicity, we assume that this object only has one degree of freedom (DOF), e.g. the vertical displacement of the centre of gravity of a floating vessel. We also assume that the object satisfies the equation of motion given by a linear mass-damping-stiffness system:

with $m$ the mass of the object, $c$ the damping and $k$ the stiffness. $F(t)$ is a time-dependent forcing term. We also provide appropriate initial conditions, in that case

In General, and mostly depending on the complexity of the forcing term $F(t)$, it is difficult to find an analytical expression that is defined at all times, $u(t)\forall t\in [0,\mathrm{\infty })$, see red blue in the following figure.

Instead, we might be interested in knowing the value of the function at specific points in time, $u(t_0),u(t_1),u(t_2),...,u(t_N)$, see red dots in the previous figure. From these set of values, one can reconstruct an approximated function ${\tilde{u}}_N(t)$ by, for instance, using a linear interpolation between points (green line in the figure).

For smooth enough functions, $u(t)$, as we increase the number of evaluation points in time, $N$, the approximated solution solution ${\tilde{u}}_N(t)$ will be closer to $u(t)$.

Notation#

In these notes we will use the following notation, see the figure below:

• $u_i:=u(t_i)$, the function evaluated at time $t_i$.

• $\mathrm{\Delta }{t}_i:=t_i-t_{i-1}$, the time step between two consecutive time steps, $t_{i-1}$ and $t_i$. When considering constant time steps, in an interval of time $t\in [0,T]$ with $N$ time steps, the time step size will be $\mathrm{\Delta }t=T/N$.