Difference schemes of optimum degree of implicitness for a family of simple ODEs with blow-up solutions

It has been speculated that the linearly implicit linearized trapezoidal rule should be especially useful for computing blow-up solutions since it features discrete blow-up. We study the question of how implicit a scheme should be from several different points of view: we discuss further the properties of the linearized trapezoidal and of its discrete blow-up, we give exact schemes for a family of ODEs with polynomial nonlinearity, and that shows ‘the optimum degree of implicitness’ they need; we show that other standard schemes (the trapezoidal rule itself and the implicit midpoint rule) are exact on certain differential equations; we compare for several schemes and different nonlinearities the size of the leading error terms; and we briefly discuss two ways of adapting the degree of implicitness of a scheme to a given differential equation.