Tensor product derivative matching for wave propagation in inhomogeneous media

Zhao, Shan; Wang, Gang

doi:10.1002/mop.20378

Cited by 6 publications

(2 citation statements)

References 22 publications

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“…al. developed the matched interface and boundary (MIB) method for interface problems in [170,171,174]. The MIB method has also been applied as a general scheme for accommodating some complex boundary conditions in high-order spatial discretization of PDEs [173,172].…”

Section: Other Related Workmentioning

confidence: 99%

An Introduction to the Immersed Boundary and the Immersed Interface Methods

Dillon¹,

Li²

2009

Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore

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This paper attempts to give a brief survey or tutorial on the immersed boundary (IB) method and the immersed interface method (IIM). The immersed boundary method was originally introduced by Peskin for studying flow patterns around heart valves and for studying blood flow in the heart [118] and has since been applied to many other problems, particularly, in biophysics. The IB method is a mathematical modeling framework as well as a numerical method. The original motivation of the immersed interface method [76,80] was to improve the accuracy of the IB method, at least from first-order accuracy to second; and deal with discontinuous coefficients in the governing equations. The IIM is a sharp interface method based on Cartesian grids or triangulation. The IIM makes use of jump conditions across interfaces so that the finite difference/element discretization can be accurate. Some new developments in the immersed interface method include the augmented 1 Interface Problems and Methods in Biological and Physical Flows Downloaded from www.worldscientific.com by Mr Yaosheng Liang on 03/20/15. For personal use only. 2 R. H. Dillon and Z. Li immersed interface method; immersed finite element method; and singularity removal techniques. The immersed boundary and the interface methods have been coupled with different evolution schemes for free boundary and moving interface problems.Keywords: Immersed boundary method, immersed interface method, interface problem, irregular domain, discontinuous coefficient, singular source, delta function, discontinuous or non-smooth solution, augmented method, immersed finite element method, source singularity removal.

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Section: Other Related Workmentioning

confidence: 99%

An Introduction to the Immersed Boundary and the Immersed Interface Methods

Dillon¹,

Li²

2009

Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore

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“…The proposed MIB method maintains the collocation feature of central FD method over the entire computational domain without resorting to an optimization procedure as that of the FLAME [16][17][18][19][20]. The MIB method is originated from the hierarchical derivative matching method [34,35], originally proposed for simulating electromagnetic wave scattering and propagation in inhomogeneous media. For solving elliptic interface problems with curved interfaces, up to sixth-order MIB schemes have been constructed [36] as a generalization of the immersed boundary method [37], immerse interface method [38,39], and ghost fluid method [40].…”

Section: Introductionmentioning

confidence: 99%

Matched interface and boundary (MIB) for the implementation of boundary conditions in high‐order central finite differences

Zhao

Wang

2008

Numerical Meth Engineering

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SUMMARYHigh-order central finite difference schemes encounter great difficulties in implementing complex boundary conditions. This paper introduces the matched interface and boundary (MIB) method as a novel boundary scheme to treat various general boundary conditions in arbitrarily high-order central finite difference schemes. To attain arbitrarily high order, the MIB method accurately extends the solution beyond the boundary by repeatedly enforcing only the original set of boundary conditions. The proposed approach is extensively validated via boundary value problems, initial-boundary value problems, eigenvalue problems, and high-order differential equations. Successful implementations are given to not only Dirichlet, Neumann, and Robin boundary conditions, but also more general ones, such as multiple boundary conditions in high-order differential equations and time-dependent boundary conditions in evolution equations. Detailed stability analysis of the MIB method is carried out. The MIB method is shown to be able to deliver high-order accuracy, while maintaining the same or similar stability conditions of the standard high-order central difference approximations. The application of the proposed MIB method to the boundary treatment of other non-standard high-order methods is also considered.

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Simplified immersed interface methods for elliptic interface problems with straight interfaces

Feng

2011

Numerical Methods Partial

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In this article, we propose simplified immersed interface methods for elliptic partial/ordinary differential equations with discontinuous coefficients across interfaces that are few isolated points in 1D, and straight lines in 2D. For one-dimensional problems or two-dimensional problems with circular interfaces, we propose a conservative second-order finite difference scheme whose coefficient matrix is symmetric and definite. For two-dimensional problems with straight interfaces, we first propose a conservative first-order finite difference scheme, then use the Richardson extrapolation technique to get a second-order method. In both cases, the finite difference coefficients are almost the same as those for regular problems. Error analysis is given along with numerical example.

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Tensor product derivative matching for wave propagation in inhomogeneous media

Cited by 6 publications

References 22 publications

An Introduction to the Immersed Boundary and the Immersed Interface Methods

An Introduction to the Immersed Boundary and the Immersed Interface Methods

Matched interface and boundary (MIB) for the implementation of boundary conditions in high‐order central finite differences

Simplified immersed interface methods for elliptic interface problems with straight interfaces

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