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

TL;DR

This paper investigates the properties of the TR-BDF2 numerical method, focusing on monotonicity, stability, and positivity, and introduces hybrid variants that improve robustness for stiff problems.

Contribution

It analyzes the absolute monotonicity of TR-BDF2, computes its monotonicity radius, and proposes hybrid variants that enhance stability and accuracy in stiff problem contexts.

Findings

Maximized the radius of absolute monotonicity for TR-BDF2.

Hybrid variants improve robustness at high CFL numbers.

Strategies outperform existing methods in stiff and mildly stiff scenarios.

Abstract

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.