DOC PREVIEW
Matched interface and boundary (MIB) for the implementation

This preview shows page 1-2-3-19-20-39-40-41 out of 41 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 41 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 41 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 41 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 41 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 41 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 41 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 41 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 41 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 41 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2009; 77:1690–1730Published online 16 September 2008 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2473Matched interface and boundary (MIB) for the implementationof boundary conditions in high-order central finite differencesShan Zhao1, ∗, †andG.W.Wei2, 31Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487, U.S.A.2Department of Mathematics, Michigan State University, East Lansing, MI 48824, U.S.A.3Department of Electrical and Computer Engineering, Michigan State University,East Lansing, MI 48824, U.S.A.SUMMARYHigh-order central finite difference schemes encounter great difficulties in implementing complex boundaryconditions. This paper introduces the matched interface and boundary (MIB) method as a novel boundaryscheme to treat various general boundary conditions in arbitrarily high-order central finite differenceschemes. To attain arbitrarily high order, the MIB method accurately extends the solution beyond theboundary by repeatedly enforcing only the original set of boundary conditions. The proposed approach isextensively 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 inhigh-order differential equations and time-dependent boundary conditions in evolution equations. Detailedstability analysis of the MIB method is carried out. The MIB method is shown to be able to deliverhigh-order accuracy, while maintaining the same or similar stability conditions of the standard high-ordercentral difference approximations. The application of the proposed MIB method to the boundary treatmentof other non-standard high-order methods is also considered. Copyright q 2008 John Wiley & Sons, Ltd.Received 7 January 2008; Revised 19 August 2008; Accepted 19 August 2008KEY WORDS: high-order methods; central finite differences; complex boundary conditions; matchedinterface and boundary1. INTRODUCTIONFinite difference (FD) method is the oldest while still a widely used approach for the numericalsolution of partial differential equations [1–4]. To achieve high-order accuracy as well as high∗Correspondence to: Shan Zhao, Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487, U.S.A.†E-mail: [email protected]/grant sponsor: NSF; contract/grant numbers: DMS-0731503, DMS-0616704Contract/grant sponsor: NSF; contract/grant numbers: IIS-0430987, DMS-0616704Contract/grant sponsor: NIH; contract/grant number: CA127189-01Copyright q 2008 John Wiley & Sons, Ltd.MATCHED INTERFACE AND BOUNDARY (MIB) 1691cost-efficiency for practical applications, numerous high-order FD methods have been developed inthe literature [5–12], including, standard, Euler sum, non-standard, compact, spectrally weighted,and optimized FD schemes, to name only a few. Typically, these high-order FD methods usewide stencils. Thus, to maintain a designed high-order accuracy, special numerical treatments arerequired near boundaries where these FD kernels may refer to nodes outside the computationaldomain. However, it is numerically challenging to construct a boundary closure method that isnot only highly accurate to maintain the designed level of accuracy, but also sufficiently robustto handle various boundary conditions arisen in practical problems, and free of non-physicalspurious solutions. Indeed, the development of such boundary closure methods has attracted muchof research attention in scientific and engineering computations.The boundary closure of high-order FD schemes with wide stencils can be carried out inessentially two ways: one is to employ the information on a small fictitious domain outside theboundary, while the other relies only on the information inside the boundary. Many different typesof boundary closure methods have been proposed in the literature in the framework of the latterone. For example, one type of method builds boundary conditions into differentiation kernels [13],so that both the differential equation and its boundary conditions can be satisfied simultaneously.However, this technique may not be robust enough to handle general boundary conditions. Inanother type, boundary conditions are imposed in the differential equation discretization by usingpenalty-like terms [14, 15]. Apart from the construction of a delicate procedure to select a penaltyfactor, the main problem of the penalty method is the possible loss of high accuracy, whichis at odds with the spirit of using high-order FD methods. If certain analytical features, suchas boundary layers and singularities, are known apriorinear the boundary, such local featurescould be included in numerical discretization to promote a more accurate simulation. To this end,the flexible local approximation method (FLAME) [16–20] can be employed, which provides ageneral framework for integrating analytical features into local FD approximations in a very simplemanner. For time-dependent problems, summation-by-parts operators have been constructed forFD approximations of first and second derivatives [21, 22]. Effective boundary closure schemesbased on the simultaneous approximation term principle have been presented to maintain bothhigh-order accuracy and stability [21–23]. The most commonly used boundary closure method forhigh-order FD approaches in this category is to employ progressively more asymmetric versions ofdifferential kernels near the boundary [24, 25]. In other words, one-sided FD (OFD) approximationsare employed near boundaries, which do not involve nodes outside the computational domain.In practice, Chebyshev-type node clustering toward the ends of the domain is usually utilized topermit high accuracy. This kind of non-uniform grid is also widely used in the spectral collocationmethod. However, using the Chebyshev-type node clustering, the grid spacing h at the boundariesis much smaller than the interior ones. Consequently, such node clustering generally induceshigh conditional numbers in solving elliptic problems and severe stability constraints in solvingtime-dependent problems.At present, it is of considerable interest to study the other type of boundary closure methods,i.e. the fictitious domain boundary method. Moreover, to avoid the difficulty associated


Matched interface and boundary (MIB) for the implementation

Download Matched interface and boundary (MIB) for the implementation
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Matched interface and boundary (MIB) for the implementation and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Matched interface and boundary (MIB) for the implementation 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?