Newmark's schemes and Isogeometric approximation of acoustic waves with absorbing boundary conditions

In recent years there has been an increased attention to the accurate simulation of wave propagation in acoustic models. Classic simulations of acoustic waves propagation in geophysics have been based on finite difference or finite element methods also including domain decomposition methods in the case of discontinuous data at the physical interfaces. In the last decades an increasing number of works have tried to improve the low-order accuracy of finite differences or finite elements by considering either spectral or spectral element methods. In this presentation we consider the numerical approximation of 2D acoustic wave problems by the Isogeometric method based on $B$-spline basis functions in space, and Newmark's finite difference schemes of different order in time, both explicit and implicit. Proper boundary conditions are considered. While homogeneous Neumann condition provides a good mathematical representation of a free surface in correspondence of which full reflection occurs, first order absorbing boundary conditions are imposed in connection with the numerical simulation of wave propagation in infinite domains, in order to truncate the original unbounded domain into a finite one and to keep spurious wave reflections as low as possible. Several numerical examples in different geometries illustrate the stability and convergence properties of the proposed numerical methods with respect to all the approximation parameters, namely the local polynomial degree of the basis function, the size of the mesh refinement, the time step and the parameters of the Newmark's time advancing scheme.