Uniformly exponentially stable approximations for a class of damped systems

By Sylvain Ervedoza, Enrique Zuazua. Journal des Mathématiques Pures et Appliquées, 2009

75 citations recorded by the source corpus.

We consider time semi-discrete approximations of a class of exponentially stable infinite-dimensional systems modeling, for instance, damped vibrations. It has recently been proved that for time semi-discrete systems, due to high frequency spurious components, the exponential decay property may be lost as the time step tends to zero. We prove that adding a suitable numerical viscosity term in the numerical scheme, one obtains approximations that are uniformly exponentially stable. This result is then combined with previous ones on space semi-discretizations to derive similar results on fully-discrete approximation schemes. Our method is mainly based on a decoupling argument of low and high frequencies, the low frequency observability property for time semi-discrete approximations of conservative linear systems and the dissipativity of the numerical viscosity on the high frequency components. Our methods also allow to deal directly with stabilization properties of fully discrete approximation schemes without numerical viscosity, under a suitable CFL type condition on the time and space discretization parameters.

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/0066ca66791bdbb9787f0fa197839e7f6d253f05

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Citations in the corpus

Reduced-order Unscented Kalman Filtering with application to parameter identification in large-dimensional systems (2011, methodology): “In other cases some carefully devised numerical viscosity may be used to recover the asymptotic stability as in [8].”
Characterization by observability inequalities of controllability and stabilization properties (2018, background): “vergence of Riccati operators for secondorder systems ¨y+ Ay= Buwith A: D(A) → Xpositive selfadjoint with compact inverse and B bounded control operator. General conservative equations are treated in [10] where it is proved that adding a viscosity term in the numerical scheme helps to recover a unif”
The Wave Equation: Control and Numerics (2012, background): “This is the idea that ha s been developed in [99] for the 1 and 2-d wave equation and then later formalized in a m uch more general form in [33, 34].”
ASYMPTOTIC BEHAVIOR OF SECOND-ORDER DISSIPATIVE EVOLUTION EQUATIONS COMBINING POTENTIAL WITH NON-POTENTIAL EFFECTS ∗ (2009, background): “In the case of linear hyperbolic PDE’s, it would be worthwhile to compare it with spectral analysis methods, see for example [25,27].”