A regularization technique for accurate reconstruction of numerical solution of wave propagation problems

In the numerical solution of wave propagation problems, spurious oscillations occur in the exact time integration of the related equation of motion. This is due to the high frequencies introduced by the spatial discretization, given the small size of the mesh elements and the integration step required to capture the wave phenomena. Currently, an answer to this problem is given by the use of the smoothing properties, intrinsic to the scheme or obtained through artificial viscosity, of dissipative time integration methods. More recently, in an alternative approach to the problem, the solution has been regularized via a post processing smoothing technique. In particular, on the basis of an initial nondissipative scheme, at a fixed observation time a series of steps of an appropriate dissipative time integration method achieves the desired smoothing. However, in both approaches, as the dissipative steps are performed, the noise progressively decreases, but the important values related to the peak regions of the solution degrade significantly. Here we describe a regularization process that automatically returns a solution where the noise has been eliminated but does not affect the significant regions of the solution. The presented technique recognizes the flat or peak shapes of the original solution among the oscillating components representing the noise. Operationally, the presented algorithm, starting from a nondissipative stepby-step scheme for time integration, iteratively smooths the related kinematic quantities and finally recovers the regularized solution as a suitable composition of smoothed and unsmoothed subdomains.