Optimal convergence analysis of the energy-preserving immersed weak Galerkin method for second-order hyperbolic interface problems in inhomogeneous media

This article reports our explorations for solving second-order hyperbolic interface problems by immersed weak Galerkin (IWG) method on interface independent meshes. The method presented here uses IWG functions for the discretization in spatial variable. The study includes the Newmark algorithm which has been used extensively in applications. The stability analysis based on the energy method is presented for semi-discrete and fully-discrete schemes under some conditions on parameters of the Newmark algorithm. In this work, we carried out a convergence analysis and obtained optimal a priori error estimates in both energy and L2 norms for the semi-discrete and fully-discrete schemes under piecewise H2 regularity assumption in space and some conditions on parameters of the Newmark algorithm. We demonstrate that the maximal error in the L2-norm error over a finite time interval converges optimally as O(h2+τr(γ)), where r(γ)=1 if γ≠1/2, r(1/2)=2, h and τ are the mesh size and the time step, respectively. Numerical examples are provided to confirm theoretical findings and illustrate the efficiency of the method for standing and traveling waves.