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Finite difference discretization with variable mesh of the Schrödinger equation in a variable domain. (English) Zbl 0745.65071

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.

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
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