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

Crank-Nicolson finite element discretizations for a two-dimenional linear Schroedinger-type equation posed in noncylindrical domain

First published in Mathematics of Computation online 2014 (84 (2015), 1571-1598), published by the American Mathematical SocietyMotivated by the paraxial narrow–angle approximation of the Helmholtz equation in domains of variable topography, we consider an initialand boundary-value problem for a general Schr¨odinger-type equation posed on a two space-dimensional noncylindrical domain with mixed boundary conditions. The problem is transformed into an equivalent one posed on a rectangular domain, and we approximate its solution by a Crank–Nicolson finite element method. For the proposed numerical method, we derive an optimal order error estimate in the L2 norm, and to support the error analysis we prove a global elliptic regularity theorem for complex elliptic boundary value problems with mixed boundary conditions. Results from numerical experiments are presented which verify the optimal order of convergence of the method

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

NanoConstruct:A Web Application Builder of Ellipsoidal Nanoparticles for the investigation of their crystal growth, their stability, and the calculation of their atomistic descriptors

NanoConstruct is a state-of-the-art computational tool that enables a) the digital construction of ellipsoidal neutral energy minimized nanoparticles (NPs) in vacuum through its graphical user-friendly interface, and b) the calculation of NPs atomistic descriptors. It allows the user to select NP’s shape and size by inserting its ellipsoidal axes and rotation angle while the NP material is selected by uploading its Crystallography Information File (CIF). To investigate the stability of materials not yet synthesised, NanoConstruct allows the substitution of the chemical elements of an already synthesized material with chemical elements that belong into the same group and neighbouring rows of the periodic table. The process is divided into three stages: 1) digital construction of the unit cell, 2) digital construction of NP using geometry rules and keeping its stoichiometry and 3) energy minimization of the geometrically constructed NP and calculation of its atomistic descriptors. In this study, NanoConstruct was applied for the investigation of the crystal growth of Zirconia (ZrO2) NPs when in the rutile form. The most stable configuration and the crystal growth route were identified, showing a preferential direction for the crystal growth of ZrO2 in its rutile form. NanoConstruct is freely available through the Enalos Cloud Platform (https://enaloscloud.novamechanics.com/riskgone/nanoconstruct/)