Backward-Euler

The backward-Euler method also truncates the Taylor series after two terms. The difference is that the derivative is evaluated at point $t+h$ instead of at point $t$.

$$y(t+h) = y(t) + hy^{(1)}(t+h)$$ (1.3)

Assuming that the value at point $t$ is correct, the backward-Euler method computes the value at point $t+h$ with a local truncation error that scales with $h^2$. The backward-Euler method always gives undershoots on the original curve.

Normally we do not know the derivative at point $t+h$, although we need it to compute the function value at point $t+h$. In practice this requires a rearrangement of the equation. We call such a numerical scheme an implicit numerical scheme. For most equations implicit schemes are more stable than explicit schemes because of the undershoots.

Figure: Graphical illustration of the backward-Euler method. To obtain point 2 from point 1, we take the derivative at point 2 and extrapolate it at point 1. To obtain point 3 starting at point 2, we do the same: take the derivative at point 3 and extrapolate it at point 2.