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Uniformly exponentially stable approximations for a class of damped systems. (English) Zbl 1163.74019
Summary: 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 on the dissipativity of numerical viscosity for 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 time and space discretization parameters.
74H15Numerical approximation of solutions for dynamical problems in solid mechanics
74H45Vibrations (dynamical problems in solid mechanics)
74S20Finite difference methods in solid mechanics