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.

##### MSC:

74H15 | Numerical approximation of solutions for dynamical problems in solid mechanics |

74H45 | Vibrations (dynamical problems in solid mechanics) |

74S20 | Finite difference methods in solid mechanics |