B11 - numerical integration of ODEs

considering an ODE:

\dot{\vec{x}} = A \vec{x} ⟹ \dot{\vec{x}} = f (\vec{x}, t)

where, $\vec{x}$ represents the system state, and $f$ may be a non linear function (vector field)

the trajectory is path $\vec{x} (t) : {\vec{x}}_{k} = \vec{x} (t_{k}) = \vec{x} (k Δ t)$ for $t \in [0, N]$
let $x_{k + 1} = F (x_{k})$
note: $f$ may be a solution to a PDE

forward euler method

\begin{matrix} \frac{d \vec{x} (t)}{d t} \approx \frac{\vec{x} (t + Δ t) - \vec{x} (t)}{Δ t} \approx f (\vec{x} (t)) \\ \vec{x} (t + Δ t) \approx \vec{x} (t) + Δ t f (\vec{x} (t)) \end{matrix}

this is the forward euler method, and can be rewritten as:

{\vec{x}}_{k + 1} = {\vec{x}}_{k} + Δ t f ({\vec{x}}_{k})

this is an explicit scheme as ${\vec{x}}_{k + 1}$ is an explicit function of ${\vec{x}}_{k}$
considering a linear ODE: $\dot{\vec{x}} = A \vec{x} :$

{\vec{x}}_{k + 1} = {\vec{x}}_{k} + Δ t A {\vec{x}}_{k} = (I + Δ t A) {\vec{x}}_{k}

where, $A$ is a matrix, and $I$ is the identity matrix

the bracketed terms are the first two terms in the taylor series of $\vec{x} = \exp (A Δ t) {\vec{x}}_{k}$

backward euler method

the backward euler method simply uses:

\begin{matrix} \frac{{\vec{x}}_{k + 1} - {\vec{x}}_{k}}{Δ t} \approx \dot{\vec{x}} (t_{k + 1}) = f ({\vec{x}}_{k + 1}) \\ ⟹ {\vec{x}}_{k + 1} = {\vec{x}}_{k} + Δ t f ({\vec{x}}_{k + 1}) \end{matrix}

this is an implicit scheme
if $\dot{\vec{x}} = A \vec{x} :$

{\vec{x}}_{k + 1} = {\vec{x}}_{k} + Δ t A {\vec{x}}_{k + 1} = (I + Δ t A)^{- 1} {\vec{x}}_{k}

motion

considering a simple damped harmonic oscillator:

\ddot{x} = - ω_{0}^{2} x - γ \dot{x}

rewriting this:

\begin{matrix} \dot{x} = v \\ \dot{v} = - γ v - ω_{0}^{2} x \end{matrix}

rewriting these simultaneous equations in matrix form:

\frac{d}{d t} [\begin{matrix} x \\ v \end{matrix}] = [\begin{matrix} 0 & 1 \\ - ω_{0}^{2} & - γ \end{matrix}] [\begin{matrix} x \\ v \end{matrix}]