Stepsize restrictions for stability in the numerical solution of ordinary and partial differential equations (Q1096345)

The authors consider one-step recursions of the form \(u_ n=\phi (\Delta tA_ n)u_{n-1}\), where \(A_ n\) are matrices and \(\phi\) a rational approximation to the exponential function. The application of Runge-Kutta or Rosenbrock methods to linear, time-dependent systems of ordinary differential equation results, of course, in recursions of this form. The paper, largely of an expository nature, presents a number of sufficient conditions for the stability of the recursion. The results are of particular relevance to the time integration of evolutionary partial differential equations and a special emphasis is given to the maximum norm.