A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem

Gibou, Frédéric; Fedkiw, Ronald

doi:10.1016/j.jcp.2004.07.018

Cited by 231 publications

(242 citation statements)

References 38 publications

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“…In [34] and [36], we extended the ghost cell method to attain second-order accuracy on interior problems (i.e., p is constant in Ω c ) with boundary conditions that depend upon the geometry (e.g., curvature) and without a jump condition on the normal derivative. A similar extension to the ghost cell method was presented in [19] to solve Laplace's equation without geometric boundary conditions and yielded fourth-order convergence on fixed domains and third-order convergence on moving boundaries. In [38], we extended our approach to solve systems like (18) in the case where h = 0 and D was constant in Ω and Ω c (with different constants).…”

Section: The Ghost Cell Methodsmentioning

confidence: 96%

“…(The diffusion constant may be discontinuous across the interface Σ; this case is treated below.) If (x i−1 , y j ), (x i , y j ), and (x i+1 , y j ) are all in the same region, i.e., (19) or (20) then we can use the standard second-order discretization (21) Suppose, however, that (x i , y j ) and (x i+1 , y j ) are not in the same region. Assume without loss of generality that (x i , y j ) ∈ Ω and (x i+1 , y j ) ∈ Ω c ; the case where (x i , y j ) ∈ Ω c and (x i+1 , y j ) ∈ Ω is treated similarly.…”

Section: Ghost Cell Extrapolations For the Diffusionalmentioning

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: 96%

Section: Ghost Cell Extrapolations For the Diffusionalmentioning

confidence: 99%

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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“…This methodology has been applied to a wide range of applications including deflagration in Fedkiw et al [43], compressible/incompressible fluids in Caiden et al [24], flame propagation in Nguyen et al [97], the Poisson equation with jump conditions in Liu et al [78], free surface flows in Enright et al [40], as well as in computer graphics [96,39]. It was developed for the Poisson and the diffusion equations on irregular domains with Dirichlet boundary conditions and their applications in Gibou et al [47,45,44,46]. In what follows, we describe the algorithms, point out common misconceptions and describe how to properly implement those methods.…”

Section: The Ghost-fluid Methods For the Diffusion And The Poisson Equmentioning

confidence: 99%

“…• In the case where third-or fourth-order accuracy is desired, the second-order central differencing used in equations (8) and (9) are replaced by the standard fourth-order accurate central differencing (see [45]). …”

Section: Remarksmentioning

confidence: 99%

High Resolution Sharp Computational Methods for Elliptic and Parabolic Problems in Complex Geometries

2012

Self Cite

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We present a review of some of the state-of-the-art numerical methods for solving the Stefan problem and the Poisson and the diffusion equations on irregular domains using (i) the level-set method for representing the (possibly moving) irregular domain's boundary, (ii) the ghost-fluid method for imposing the Dirichlet boundary condition at the irregular domain's boundary and (iii) a quadtree/octree nodebased adaptive mesh refinement for capturing small length scales while significantly reducing the memory and CPU footprint. In addition, we highlight common misconceptions and describe how to properly implement these methods. Numerical experiments illustrate quantitative and qualitative results.

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“…We now describe a second-order accurate extension of the ghost fluid method (Fedkiw et al, 1999;Liu et al, 2000;Gibou et al, 2002Gibou et al, , 2003Gibou and Fedkiw, 2005) to solve the Poisson-like system…”

Section: A Improvements To the Ghost Fluid Methodsmentioning

confidence: 99%

Nonlinear simulation of the effect of microenvironment on tumor growth

Macklin

Lowengrub

2007

Journal of Theoretical Biology

176

190

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In this paper, we present and investigate a model for solid tumor growth that incorporates features of the tumor microenvironment. Using analysis and nonlinear numerical simulations, we explore the effects of the interaction between the genetic characteristics of the tumor and the tumor microenvironment on the resulting tumor progression and morphology. We find that the range of morphological responses can be placed in three categories that depend primarily upon the tumor microenvironment: tissue invasion via fragmentation due to a hypoxic microenvironment; fingering, invasive growth into nutrient-rich, biomechanically unresponsive tissue; and compact growth into nutrient-rich, biomechanically responsive tissue. We found that the qualitative behavior of the tumor morphologies was similar across a broad range of parameters that govern the tumor genetic characteristics. Our findings demonstrate the importance of the impact of microenvironment on tumor growth and morphology and have important implications for cancer therapy. In particular, if a treatment impairs nutrient transport in the external tissue (e.g., by anti-angiogenic therapy), increased tumor fragmentation may result, and therapy-induced changes to the biomechanical properties of the tumor or the microenvironment (e.g., antiinvasion therapy) may push the tumor in or out of the invasive fingering regime.

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A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem

Cited by 231 publications

References 38 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 Resolution Sharp Computational Methods for Elliptic and Parabolic Problems in Complex Geometries

Nonlinear simulation of the effect of microenvironment on tumor growth

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