Monotonicity, positivity and strong stability of the TR-BDF2 method and of its SSP extensions

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.