In this work we study the stability regions of linear multistep or multiderivative multistep methods for initial-value problems by using techniques that are straightforward to implement in modern computer algebra systems. In many applications, one is interested in (i) checking whether a given subset of the complex plane (e.g. a sector, disk, or parabola) is included in the stability region of the numerical method, (ii) finding the largest subset of a certain shape contained in the stability region of a given method, or (iii) finding the numerical method in a parametric family of multistep methods whose stability region contains the largest subset of a given shape. First we describe a simple procedure to exactly calculate the stability angle $α$ in the definition of $A(α)$-stability. As an illustration, we consider two finite families of implicit multistep methods: we exactly compute the stability angles for the $k$-step BDF methods ($3\le k\le 6$) and for the $k$-step second-derivative multistep methods of Enright ($3\le k\le 7$). Next we determine the exact value of the stability radius in the BDF family for each $3\le k\le 6$, that is, the radius of the largest disk in the left half of the complex plane, symmetric with respect to the real axis, touching the imaginary axis and lying in the stability region of the corresponding method. Finally, we demonstrate how some Schur--Cohn-type theorems of recursive nature and not relying on the RLC method can be used to exactly solve some optimization problems within infinite parametric families of multistep methods. As an example, we choose a two-parameter family of implicit-explicit (IMEX) methods: we identify the unique method having the largest stability angle in the family, then we find the unique method in the same family whose stability region contains the largest parabola.
King Abdullah University of Science & Technology（阿卜杜拉国王科技大学）；Széchenyi István University（塞切尼伊什特万大学）；University of Massachusetts Dartmouth（马萨诸塞大学达特茅斯分校）
Stability analysis of consistent splitting implicit-explicit multistep methods up to ninth-order accuracy for incompressible flows
School of Mathematics, Nanjing University of Aeronautics and Astronautics（南京航空航天大学数学学院）；Key Laboratory of Mathematical Modeling and High Performance Computing of Air Vehicles (NUAA), MIIT（航空飞行器数学建模与高性能计算教育部重点实验室（南航））；School of Mathematics, Nanjing Audit University（南京审计大学数学学院）
A semi-generating function approach to the stability of implicit-explicit multistep methods for nonlinear parabolic equations
Nanjing University of Aeronautics and Astronautics（南京航空航天大学）；Key Laboratory of Mathematical Modeling and High Performance Computing of Air Vehicles (NUAA), MIIT（航空飞行器数学建模与高性能计算重点实验室（工信部））；The Chinese University of Hong Kong (Shenzhen)（香港中文大学（深圳））；Shenzhen International Center for Industrial and Applied Mathematics, Shenzhen Research Institute of Big Data（深圳市大数据研究院工业与应用数学国际中心）；Guangzhou Nanfang College（广州南方学院）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
Exact optimal values of step-size coefficients for boundedness of linear multistep methods
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；Eötvös Loránd University（厄特沃什·罗兰大学）；Budapest University of Technology and Economics（布达佩斯技术与经济大学）
