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Finite difference schemes on unbounded domains. (English) Zbl 1085.65076

Mickens, Ronald E. (ed.), Advances in the applications of nonstandard finite difference schemes. Hackensack, NJ: World Scientific (ISBN 981-256-404-7/hbk). 343-384 (2005).
Summary: We discuss the nonstandard problem of using the finite difference method to solve numerically a partial differential equation posed on an unbounded domain. We propose different strategies to construct so-called discrete artificial boundary conditions (ABCs) and present an efficient implementation by the sum-of-exponential ansatz. The derivation of the ABCs is based on the knowledge of the exact solution, the construction of asymptotic solutions or the usage of a continued fraction expansion to a second-order difference equation.
Our approach is explained by means of three different types of partial differential equations arising in option pricing, in quantum mechanics and in (underwater) acoustics. Finally, we conclude with an illustrating numerical example from underwater acoustics showing the superiority of our new approach.
For the entire collection see [Zbl 1079.65005].

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K05 Heat equation
35K15 Initial value problems for second-order parabolic equations
76Q05 Hydro- and aero-acoustics
76M20 Finite difference methods applied to problems in fluid mechanics
91G80 Financial applications of other theories
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