High-Order Embedded Finite Difference Schemes for Initial Boundary Value Problems on Time Dependent Irregular Domains

This paper considers a family of spatially discrete approximations, including boundary treatment, to initial boundary value problems in evolving bounded domains. The presented method is based on the Cartesian grid embedded Finite-Difference method, which was initially introduced by Abarbanel and Dit...

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Bibliographic Details
Published in:Journal of scientific computing Vol. 39; no. 3; pp. 394 - 440
Main Authors: Ditkowski, Adi, Harness, Yuval
Format: Journal Article
Language:English
Published: Boston Springer US 01.06.2009
Springer Nature B.V
Subjects:
Algorithms
Approximation
Boundaries
Boundary conditions
Boundary value problems
Computational Mathematics and Numerical Analysis
Dirichlet problem
Finite difference method
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematical models
Mathematics
Mathematics and Statistics
Solidification
Theoretical
Time dependence
Embedded finite difference
Diffusion equation
Complex geometry
Heat equation
Initial boundary value problems
Moving boundaries
ISSN:0885-7474, 1573-7691
Online Access:Get full text
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Summary:This paper considers a family of spatially discrete approximations, including boundary treatment, to initial boundary value problems in evolving bounded domains. The presented method is based on the Cartesian grid embedded Finite-Difference method, which was initially introduced by Abarbanel and Ditkowski (ICASE Report No. 96-8, 1996 ; and J. Comput. Phys. 133(2), 1997 ) and Ditkowski (Ph.D. thesis, Tel Aviv University, 1997 ), for initial boundary value problems on constant irregular domains. We perform a comprehensive theoretical analysis of the numerical issues, which arise when dealing with domains, whose boundaries evolve smoothly in the spatial domain as a function of time. In this class of problems the moving boundaries are impenetrable with either Dirichlet or Neumann boundary conditions, and should not be confused with the class of moving interface problems such as multiple phase flow, solidification, and the Stefan problem. Unlike other similar works on this class of problems, the resulting method is not restricted to domains of up to 3-D, can achieve higher than 2nd-order accuracy both in time and space, and is strictly stable in semi-discrete settings. The strict stability property of the method also implies, that the numerical solution remains consistent and valid for a long integration time. A complete convergence analysis is carried in semi-discrete settings, including a detailed analysis for the implementation of the diffusion equation. Numerical solutions of the diffusion equation, using the method for a 2nd and a 4th-order of accuracy are carried out in one dimension and two dimensions respectively, which demonstrates the efficacy of the method.
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ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-009-9277-1