  We analyze the one-step method TR-BDF2 from the point of view of
monotonicity, strong stability and positivity. All these properties are
strongly related and reviewed in the common framework of absolute monotonicity.
The radius of absolute monotonicity is computed and it is shown that the
parameter value which makes the method L-stable is also the value which
maximizes the radius of monotonicity. Two hybrid variants of TR-BDF2 are
proposed, that reduce the formal order of accuracy and maximize the absolute
monotonicity radius, while keeping the native L-stability useful in stiff
problems. Numerical experiments compare these different hybridization
strategies with other methods commonly used in the presence of stiff and mildly
stiff source terms. The results show that both strategies provide a good
compromise between accuracy and robustness at high CFL numbers, without
suffering from the limitations of alternative approaches already available in
literature.
