Numerical Observers with Vanishing Viscosity for the 1d Wave Equation

We consider a numerical scheme associated with the iterative method developed in Ramdani et al. (ESAIM Control Optim. Calc. Var. 13(3):503–527, 2007) to recover initial conditions of conservative systems. In this method, the initial conditions are reconstructed by using observers. Here we use a finite-difference discretization in space of these observers and our aim is to prove estimates of the errors with respect to the mesh size and to the number of steps in the iterative method. This is done in the particular example of the 1d wave equation. In order to avoid restrictions of the number of steps with respect to the mesh size, we add a numerical viscosity in the numerical observers. A generalization for other equations is also given. © Springer Science+BusinessMedia New York 2014.