7,290 research outputs found
The Bohl spectrum for nonautonomous differential equations
We develop the Bohl spectrum for nonautonomous linear differential equation
on a half line, which is a spectral concept that lies between the Lyapunov and
the Sacker--Sell spectrum. We prove that the Bohl spectrum is given by the
union of finitely many intervals, and we show by means of an explicit example
that the Bohl spectrum does not coincide with the Sacker--Sell spectrum in
general. We demonstrate for this example that any higher-order nonlinear
perturbation is exponentially stable, although this not evident from the
Sacker--Sell spectrum. We also analyze in detail situations in which the Bohl
spectrum is identical to the Sacker-Sell spectrum
Multigrid Waveform Relaxation on Spatial Finite Element Meshes: The Discrete-Time Case
The efficiency of numerically solving time-dependent partial differential equations on parallel computers can be greatly improved by computing the solution on many time levels simultaneously. The theoretical properties of one such method, namely the discrete-time multigrid waveform relaxation method, are investigated for systems of ordinary differential equations obtained by spatial finite-element discretisation of linear parabolic initial-boundary value problems. The results are compared to the corresponding continuous-time results. The theory is illustrated for a one-dimensional and a two-dimensional model problem and checked against results obtained by numerical experiments
- …