A finite-difference treatment of interface conditions for the parabolic wave equation: The horizontal interface

Abstract

An elegant finite-difference technique applicable to the parabolic wave equation is developed to handle the boundary conditions at the interface between two media with different sound speeds and densities. Continuity of pressure and continuity of the normal component of particle velocity are preserved at the interface. The method is designed to be implicit and is unconditionally stable. A complete mathematical treatment for the case of a horizontal interface is described.

Finite Difference Discretizations of Some Initial and Boundary Value Problems with Interface

“…In fact, our main motivation for the present work was to prove an error estimate of the type (1.2) for the McDaniel-Lee difference scheme (cf. [7]) for which numerical experiments clearly indicate second-order rate of convergence in x and t. (Actually, our method is slightly different from the one in [7] in that we use a different evaluation of the coefficient ß(x, t) to render the scheme conservative. However, the proof of optimal-order convergence for the scheme of [7] is very similar to the one for the scheme at hand.)…”

“…[7]) for which numerical experiments clearly indicate second-order rate of convergence in x and t. (Actually, our method is slightly different from the one in [7] in that we use a different evaluation of the coefficient ß(x, t) to render the scheme conservative. However, the proof of optimal-order convergence for the scheme of [7] is very similar to the one for the scheme at hand.) The proof of ( 1.2) in the present case of the Schrödinger equation follows broadly the lines of the analogous proof for the heat equation and uses energy techniques similar to the ones that we used for Galerkin methods in [1].…”

“…Using the notation of (1.1), we seek a complex-valued u = u(x, t) for (x, t) 6 (For simplicity, we have assumed homogeneous Dirichlet boundary conditions at the endpoints; Neumann or mixed, and also nonhomogeneous, boundary conditions can be analyzed as well, with no additional conceptual complications.) McDaniel and Lee (cf., e.g., [6,7]) have studied implicit, CrankNicolson-type finite difference approximations for this problem and used them in computations. In fact, our main motivation for the present work was to prove an error estimate of the type (1.2) for the McDaniel-Lee difference scheme (cf.…”

“…}, 0 < j < J + 1, continuous on [0,1] and vanishing at the endpoints 0 and 1. Endowing Sh with the usual hat function basis {<p }, 1 < j < J, one obtains a standard finite element approximation wh e Sh to w , given by 7 7 , (2 …”

A numerical treatment of the fluid/elastic interface under range-dependent environments

“…In the elastic medium two sound speeds occur, the speed of P-wave cD and the speed of S-wave c S . Initial field values at r° are used along with surface points (Al)n, (A,)n+ 1 and interface boundary points 0Al0 (B )j are used as two "surface" points to solve the system of two parabolic equations in the elastic medium.…”