High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources

Zhou, Y. C.; Zhao, Shan; Feig, Michael; Wang, Gang

doi:10.1016/j.jcp.2005.07.022

Cited by 326 publications

(268 citation statements)

References 52 publications

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“…Alternative approaches for overcoming the tangential smearing problem include the immersed interface method (IIM) [30] and the matched interface boundary (MIB) method [54,57,58]. The IIM requires the use of local coordinates (based on the normal and tangential directions of the interface) to properly discretize the normal derivative jump.…”

Section: The Ghost Cell Methodsmentioning

confidence: 99%

“…We calculated the order of convergence (of the level set function ϕ) at times t ∈ {0.10, 0.20, 0.30, 0.40, 0.50} according to (58). Due to the small time step, the time error is small and is consequently dominated by the spatial error.…”

Section: Convergence Of the Overall Methodsmentioning

confidence: 99%

“…If the exact solution is unknown, then we solve on meshes with mesh lengths h 1 = h, , and , and we compute the order of convergence via (58) where each norm is computed on the common grid points. In our work, we use the discrete maximum ℓ ∞ norm for all convergence testing.…”

Section: Convergence Of the Numerical Techniquesmentioning

confidence: 99%

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A New Ghost Cell/Level Set Method for Moving Boundary Problems: Application to Tumor Growth

Macklin

Lowengrub

2008

J Sci Comput

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In this paper, we present a ghost cell/level set method for the evolution of interfaces whose normal velocity depend upon the solutions of linear and nonlinear quasi-steady reaction-diffusion equations with curvature-dependent boundary conditions. Our technique includes a ghost cell method that accurately discretizes normal derivative jump boundary conditions without smearing jumps in the tangential derivative; a new iterative method for solving linear and nonlinear quasisteady reaction-diffusion equations; an adaptive discretization to compute the curvature and normal vectors; and a new discrete approximation to the Heaviside function. We present numerical examples that demonstrate better than 1.5-order convergence for problems where traditional ghost cell methods either fail to converge or attain at best sub-linear accuracy. We apply our techniques to a model of tumor growth in complex, heterogeneous tissues that consists of a nonlinear nutrient equation and a pressure equation with geometry-dependent jump boundary conditions. We simulate the growth of glioblastoma (an aggressive brain tumor) into a large, 1 cm square of brain tissue that includes heterogeneous nutrient delivery and varied biomechanical characteristics (white matter, gray matter, cerebrospinal fluid, and bone), and we observe growth morphologies that are highly dependent upon the variations of the tissue characteristics-an effect observed in real tumor growth.

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Section: The Ghost Cell Methodsmentioning

confidence: 99%

Section: Convergence Of the Overall Methodsmentioning

confidence: 99%

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A New Ghost Cell/Level Set Method for Moving Boundary Problems: Application to Tumor Growth

Macklin

Lowengrub

2008

J Sci Comput

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“…It is very difficult to construct second-order convergent methods for this equation in the biomolecular context due to the geometric complexity, complex interface, singular charge sources and geometric singularities [15, 24, 67]. In this work, we make use of the second-order convergent matched interface and boundary (MIB) method [15, 68, 69, 73, 77, 78] to solve Eq. (60).…”

Section: Applicationsmentioning

confidence: 99%

High-order fractional partial differential equation transform for molecular surface construction

Chen

Wei

2012

Computational and Mathematical Biophysics

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Fractional derivative or fractional calculus plays a significant role in theoretical modeling of scientific and engineering problems. However, only relatively low order fractional derivatives are used at present. In general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives. This work introduces arbitrarily high-order fractional partial differential equations (PDEs) to describe fractional hyperdiffusions. The fractional PDEs are constructed via fractional variational principle. A fast fractional Fourier transform (FFFT) is proposed to numerically integrate the high-order fractional PDEs so as to avoid stringent stability constraints in solving high-order evolution PDEs. The proposed high-order fractional PDEs are applied to the surface generation of proteins. We first validate the proposed method with a variety of test examples in two and three-dimensional settings. The impact of high-order fractional derivatives to surface analysis is examined. We also construct fractional PDE transform based on arbitrarily high-order fractional PDEs. We demonstrate that the use of arbitrarily high-order derivatives gives rise to time-frequency localization, the control of the spectral distribution, and the regulation of the spatial resolution in the fractional PDE transform. Consequently, the fractional PDE transform enables the mode decomposition of images, signals, and surfaces. The effect of the propagation time on the quality of resulting molecular surfaces is also studied. Computational efficiency of the present surface generation method is compared with the MSMS approach in Cartesian representation. We further validate the present method by examining some benchmark indicators of macromolecular surfaces, i.e., surface area, surface enclosed volume, surface electrostatic potential and solvation free energy. Extensive numerical experiments and comparison with an established surface model indicate that the proposed high-order fractional PDEs are robust, stable and efficient for biomolecular surface generation.

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“…Aforementioned methods have found much success in scientific and engineering applications [6][7][8]15,18,20,[25][26][27][28]30,32,34,39,41,40,42,53,54,[57][58][59]. A possible further direction in the field could be the development of higher order interface methods [20,60,61] which are particularly desirable for problems involving both material interfaces and high frequency oscillations, such as the interaction of turbulence and shock, and high frequency wave propagation in inhomogeneous media [5].…”

Section: Introductionmentioning

confidence: 99%

Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces

Zhou

Guo

2007

Journal of Computational Physics

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169

182

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Elliptic problems with sharp-edged interfaces, thin-layered interfaces and interfaces that intersect with geometric boundary, are notoriously challenging to existing numerical methods, particularly when the solution is highly oscillatory. This work generalizes the matched interface and boundary (MIB) method previously designed for solving elliptic problems with curved interfaces to the aforementioned problems. We classify these problems into five distinct topological relations involving the interfaces and the Cartesian mesh lines. Flexible strategies are developed to systematically extends the computational domains near the interface so that the standard central finite difference scheme can be applied without the loss of accuracy. Fictitious values on the extended domains are determined by enforcing the physical jump conditions on the interface according to the local topology of the irregular point. The concepts of primary and secondary fictitious values are introduced to deal with sharp-edged interfaces. For corner singularity or tip singularity, an appropriate polynomial is multiplied to the solution to remove the singularity. Extensive numerical experiments confirm the designed second order convergence of the proposed method.

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High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources

Cited by 326 publications

References 52 publications

A New Ghost Cell/Level Set Method for Moving Boundary Problems: Application to Tumor Growth

A New Ghost Cell/Level Set Method for Moving Boundary Problems: Application to Tumor Growth

High-order fractional partial differential equation transform for molecular surface construction

Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces

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