A finite difference method for the wide‐angle "parabolic" equation in a waveguide with downsloping bottom. Issue 4 (27th December 2012)

• Record Type: Journal Article
• Authors: Antonopoulou, Dimitra C.; Dougalis, Vassilios A.; Zouraris, Georgios E.
• Is Part Of: Numerical methods for partial differential equations. Volume 29:Issue 4(2013:Jul.)
• Page Start: 1416
• Page End: 1440
• Publication Date: 2012-12-27
• Subjects: Differential equations, Partial -- Numerical solutions -- Periodicals; 515.353
• DOI: 10.1002/num.21762
• Languages: English
• ISSNs: 0749-159X

Abstract

<abstract abstract-type="main" xml:lang="en"> <title>Abstract</title> <p>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 article, we concentrate on downsloping bottom profiles and propose an additional boundary condition that yields a well‐posed problem, in fact making it <italic>L</italic>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. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013</p> </abstract>