Euler's method for concave and monotonic increasing functions?

Question

Will Euler's method (implicit or explicit) always approximate a function on or above the real solution, if the function is concave and monotonous increasing? It looks like that in my examples (all separable differential equations), and it makes sense, but I cannot find proof for it.

Answer 1

The line from $(x_n,y_n)$ to $(x_{n+1},y_{n+1})$ is always a tangent to an exact solution,

• for the explicit method it is the tangent to the exact solution through $(x_n,y_n)$,
• for the implicit method it is the tangent to the exact solution through $(x_{n+1},y_{n+1})$.

If the whole family of exact solutions consists of concave functions, then the tangent lines are always above the exact solution they are tangent to. This now means that the series of iteration points

• of the explicit method moves upwards in the family of exact solutions, and
• of the implicit method moves downwards.

This means that the explicit method gives a sequence above the exact solution and the implicit method below. Convergence order one implies the convergence towards the exact solution with the step size getting smaller.