Semi-implicit methods for differential systems with semi-stable equilibria

A stability study for long-time integrations with semi-implicit methods (or Rosenbrock-type methods) on differential systems possessing semi-stable equilibria is carried out. Here, previous results obtained for Runge–Kutta methods [Numer. Math. 97 (2004) 473] are extended to the class of semi-implicit methods. As a main result, strong A-stability turns out to be a sufficient condition to guarantee stable long-time integrations. Numerical experiments with the Robertson problem seem to confirm that this property is also necessary.