Energy stable numerical methods for hyperbolic partial differential equations using overlapping domain decomposition

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• 作者：Adam Reichert^a ; ^{areiche2@illinois.edu} ; Michael T. Heath^a ; ^{heath@illinois.edu} ; Daniel J. Bodony^b ; ^{bodony@illinois.edu}
• 关键词：High order finite difference methods ; Overlapping domain decomposition ; Numerical stability ; Generalized summation-by-parts
• 刊名：Journal of Computational Physics
• 出版年：2012
• 期刊代码：136_00219991
• 类别：cp
• 出版时间：20 June, 2012
• 卷：231
• 期：16
• 页码：5243-5265
• 文件大小：691 K

摘要

Overlapping domain decomposition methods, otherwise known as overset grid or chimera methods, are useful for simplifying the discretization of partial differential equations in or around complex geometries. Though in wide use, such methods are prone to numerical instability unless numerical diffusion or some other form of regularization is used, especially for higher-order methods. To address this shortcoming, high-order, provably energy stable, overlapping domain decomposition methods are derived for hyperbolic initial boundary value problems. The overlap is treated by splitting the domain into pieces and using generalized summation-by-parts derivative operators and polynomial interpolation. New implicit and explicit operators are derived that do not require regularization for stability in the linear limit. Applications to linear and nonlinear problems in one and two dimensions are presented, where it is found the explicit operators are preferred to the implicit ones.