This paper deals with the numerical solution of large systems of ordinary differential equations with both stiff and nonstiff parts using implicit-explicit linear multistep methods, i.e., combinations of two methods. It is assumed that the individual implicit method is \(A\)-stable. Stability properties of the implicit-explicit schemes for the scalar test equation \(w'(t)= \lambda w(t)+ \mu w(t)\) are investigated, where \(\lambda\) and \(\mu\) represent the eigenvalues of the nonstiff and stiff part, respectively.
The following questions are discussed: stability under the assumption that \(\lambda\) lies in the stability region of the individual explicit method, and restrictions on \(\lambda\) for having stability for arbitrary \(\mu\) in the left half-plane. Several popular second-order methods are studied.