A Finite Difference method for the Wide-Angle `Parabolic' equation in a waveguide with downsloping bottom

We consider the third-order wide-angle `parabolic' equation of underwater acoustics in a cylindrically symmetric fluid medium over a bottom of range-dependent bathymetry. It is known that the initial-boundary-value problem for this equation may not be well posed in the case of (smooth) bottom profiles of arbitrary shape if it is just posed e.g. with a homogeneous Dirichlet bottom boundary condition. In this paper we concentrate on downsloping bottom profiles and propose an additional boundary condition that yields a well posed problem, in fact making it $L^2$-conservative in the case of appropriate real parameters. We solve the problem numerically by a Crank-Nicolson-type finite difference scheme, which is proved to be unconditionally stable and second-order accurate, and simulates accurately realistic underwater acoustic problems.Comment: 2 figure

ADAPTIVE WEAK APPROXIMATION OF DIFFUSIONS WITH JUMPS

This work develops adaptive time stepping algorithms for the approximation of a functional of a diffusion with jumps based on a jump augmented Monte Carlo Euler–Maruyama method, which achieve a prescribed precision. The main result is the derivation of new expansions for the time discretization error, with computable leading order term in a posteriori form, which are based on stochastic flows and discrete dual backward functions. Combined with proper estimation of the statistical error, they lead to efficient and accurate computation of global error estimates, extending the results by A. Szepessy, R. Tempone, and G. E. Zouraris [Comm. Pure Appl. Math., 54 (2001), pp. 1169–1214]. Adaptive algorithms for either deterministic or trajectory-dependent time stepping are proposed. Numerical examples show the performance of the proposed error approximations and the adaptive schemes

GALERKIN METHODS FOR PARABOLIC AND SCHRODINGER EQUATIONS WITH DYNAMICAL BOUNDARY CONDITIONS AND APPLICATIONS TO UNDERWATER ACOUSTICS

In this paper we consider Galerkin-finite element methods that approximate the solutions of initial-boundary-value problems in one space dimension for parabolic and Schrodinger evolution equations with dynamical boundary conditions. Error estimates of optimal rates of convergence in L-2 and H-1 are proved for the associated semidiscrete and fully discrete Crank-Nicolson-Galerkin approximations. The problem involving the Schrodinger equation is motivated by considering the standard “parabolic” (paraxial) approximation to the Helmholtz equation, used in underwater acoustics to model long-range sound propagation in the sea, in the specific case of a domain with a rigid bottom of variable topography. This model is contrasted with alternative ones that avoid the dynamical bottom boundary condition and are shown to yield qualitatively better approximations. In the (real) parabolic case, numerical approximations are considered for dynamical boundary conditions of reactive and dissipative type