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2.2.1 Simple ODE: discretization

The simple ordinary differential equation (ODE)

\frac{\mathrm{d} y}{\mathrm{d} t} = - a   y
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with initial value
y \left( t_0 \right) = y_0
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is of first order and linear with constant coefficient \bgroup\color{DEFcolor}$a$\egroup. It has the obvious solution
y \left( t \right) = y_0   \mathrm{e}^{- a \left( t - t_0 \right)}
\enspace .
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For discrete time-steps
t^{n} = \Delta t  n + t_0
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the straight-forward replacement \bgroup\color{DEFcolor}$\mathrm{d} t \rightarrow \Delta t$\egroup gives the most simple discretization (explicit Euler scheme: approximation of \bgroup\color{DEFcolor}$y$\egroup by a piecewise linear curve)
y^{n+1} = y^{n} - a   y^{n}   \Delta t
\enspace .
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