Spectral approximation of time windows in the solution of dissipative linear differential equations

We establish a relation between the length T of the integration window of a linear differential equation x′ + Ax = b and a spectral parameter s*. This parameter is determined by comparing the exact solution x(T) at the end of the integration window to the solution of a linear system obtained from the Laplace transform of the differential equation by freezing the system matrix. We propose a method to integrate the relation s* = s*(T) into the determination of the interval of rapid convergence of waveform relaxation iterations. The method is illustrated with a few numerical examples.