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High Order Finite Difference Discretization for Composite Grid Hierarchy and Its Applications

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems General theory of linear operators Elliptic equations and systems Partial differential equations, boundary value problems

Published online by Cambridge University Press:  23 November 2015

Qun Gu ,
Weiguo Gao  and
Carlos J. García-Cervera
Qun Gu
Affiliation:
School of Mathematical Sciences, Fudan University, Shanghai 200433, P.R. China
Weiguo Gao*
Affiliation:
MOE Key Laboratory of Computational Physical Sciences and School of Mathematical Sciences, Fudan University, Shanghai 200433, P.R. China
Carlos J. García-Cervera
Affiliation:
Department of Mathematics, University of California, Santa Barbara, CA 93106, USA
*
*Corresponding author. Email addresses: 081018018@f udan.edu.cn (Q. Gu), wggao@fudan.edu.cn (W. Gao), cgarcia@math.ucsb.edu (C.J.García-Cervera)
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Abstract

We introduce efficient approaches to construct high order finite difference discretizations for solving partial differential equations, based on a composite grid hierarchy. We introduce a modification of the traditional point clustering algorithm, obtained by adding restrictive parameters that control the minimal patch length and the size of the buffer zone. As a result, a reduction in the number of interfacial cells is observed. Based on a reasonable geometric grid setting, we discuss a general approach for the construction of stencils in a composite grid environment. The straightforward approach leads to an ill-posed problem. In our approach we regularize this problem, and transform it into solving a symmetric system of linear of equations. Finally, a stencil repository has been designed to further reduce computational overhead. The effectiveness of the discretizations is illustrated by numerical experiments on second order elliptic differential equations.

Keywords

High orderpoint clustering algorithmfinite difference methodcomposite gridstencilregularization

MSC classification

Secondary: 35J57: Boundary value problems for second-order elliptic systems 47A52: Ill-posed problems, regularization 65M50: Mesh generation and refinement 65N06: Finite difference methods
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 5 , November 2015 , pp. 1211 - 1233
Copyright
Copyright © Global-Science Press 2015 

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