Finite difference discretization with variable mesh of the Schrodinger equation in a variable domain

Abstract. We consider a partial differential equation of Schrodinger type, known as the ‘parabolic’ approximation to the Helmholtz equation in the theory of sound propagation in an underwater, rangeand depth-dependent environment with a variable bottom. We solve an associated initialand boundary-value problem by a finite difference scheme of Crank-Nicolson type on a variable mesh. We prove that the method is stable in l2, establish optimal, second-order error estimates and show results of relevant numerical experiments.