Galerkin methods for the ‘Parabolic Equation’ Dirichlet problem in a variable 2-D and 3-D topography

The problem analyzed in this paper is a model for the Narrow Angle parabolic approximation of Helmholtz equation in environments in Rn, n=2,3, of variable topography used in underwater acoustics. By applying a horizontal bottom transformation combined with an exponential one, we present this Schrödinger-type Dirichlet initial and boundary-value problem in a weak formulation and prove the uniqueness of weak solution. Further, we construct Galerkin semidiscrete and Crank–Nicolson fully discrete schemes. We prove stability of numerical solution, analyze the error and prove estimates of optimal order in the L2-norm. For the 2-D case, we numerically verify the optimal order of accuracy and present numerical results for some standard Benchmark acoustical problems.