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#### Backward-Euler

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

| (1.3)

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

Normally we do not know the derivative at point , although we need it to compute the function value at point . 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.

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