What are the reason's to choose between the explicit midpoint method and the improved Euler method in solving an ordinary differential equation numerically?
Explicit midpoint method: $y_{i+1} = y_i + h f\left( x_i + \frac h 2, y_i + \frac h 2 f(x_i,y_i) \right)$
Improved Euler method: $y_{i+1} = y_i + \frac h 2 f\left( x_i, y_i \right) + \frac h 2 f\left( x_{i+1}, y_i + h f(x_i,y_i) \right)$
Both methods are of second order, so there should not be much difference in theory. What else can be said about these approaches?