On overcoming Dahlquist’s second barrier forA-stable linear multistep methods

Dahlquist’s second barrier limits the order of A-stable linear multistep methods to at most two, posing significant challenges for achieving higher accuracy in the numerical solution of stiff ordinary differential equations. Leveraging various successful techniques, many efforts have been made to develop efficient methods that overcome this fundamental obstacle through different approaches. In this paper, we survey these techniques and analyze their impact on enhancing the stability and accuracy of the resulting methods. A comprehensive understanding of these advances can assist researchers in designing more effective algorithms for stiff problems.