Abstract: The subject of the paper is the study of several nonreflecting and reflecting boundary conditions for the evolution Galerkin methods (EG) which are applied for the two-dimensional wave equation system. Different known tools are used to achieve this aim. Namely, the method of characteristics, the method of extrapolation, the Laplace transformation method, and the perfectly matched layer (PML) method. We show that the absorbing boundary conditions which are based on the use of the Laplace transformation lead to the Engquist-Majda first and second order absorbing boundary conditions, see . Further, following B\'erenger  we consider the PML method. We discretize the wave equation system with the leap-frog scheme inside the PML while the evolution Galerkin schemes are used inside the computational domain. Numerical tests demonstrate that this method produces much less unphysical reflected waves as well as the best results in comparison with other techniques studied in the paper.
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