Akrivis, Georgios D.; Dougalis, Vassilios A. Finite difference discretization with variable mesh of the Schrödinger equation in a variable domain. (English) Zbl 0745.65071 Bull. Greek Math. Soc. 31, 19-28 (1990). The authors consider the solution of the partial differential equation \(u_ r=i\alpha u_{zz}+i\beta(z,r)u\) which is used as a model in long- range, low-frequency underwater acoustics, over a domain bounded by a curved bottom. They develop a Crank-Nicolson finite difference solution, and show that it is stable, and show that there is second-order convergence. Numerical results are given for some problems, together with error estimates. Reviewer: Ll.G.Chambers (Bangor) Cited in 4 Documents MSC: 65Z05 Applications to the sciences 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 35K15 Initial value problems for second-order parabolic equations 35Q55 NLS equations (nonlinear Schrödinger equations) 35Q35 PDEs in connection with fluid mechanics 76Q05 Hydro- and aero-acoustics Keywords:variable mesh; Schrödinger equation; variable domain; low-frequency underwater acoustics; Crank-Nicolson finite difference solution; second- order convergence; numerical results; error estimates; long range PDFBibTeX XMLCite \textit{G. D. Akrivis} and \textit{V. A. Dougalis}, Bull. Greek Math. Soc. 31, 19--28 (1990; Zbl 0745.65071) Full Text: EuDML