Q1. What are the contributions mentioned in the paper "A cartesian grid embedded boundary method for the heat equation on irregular domains" ?
The authors present an algorithm for solving the heat equation on irregular time-dependent domains.
A Cartesian grid embedded boundary method for the heat equation on irregular domains
13 Nov 2001-Journal of Computational Physics (Academic Press Professional, Inc.)-Vol. 173, Iss: 2, pp 620-635
TL;DR: An algorithm for solving the heat equation on irregular time-dependent domains is presented, based on the Cartesian grid embedded boundary algorithm of Johansen and Colella, combined with a second-order accurate discretization of the time derivative.
About: This article is published in Journal of Computational Physics.The article was published on 2001-11-13 and is currently open access. It has received 161 citations till now. The article focuses on the topics: Mixed boundary condition & Boundary (topology).
Summary (1 min read)
1. INTRODUCTION
- For the time discretization, for the fixed-boundary problem the authors use either the Crank-Nicolson method or the method of Twizell, Gumel and Arigu (TGA) [10] .
- The authors algorithm is stable and achieves second-order accuracy both on problems with fixed domain and on problems with a time-dependent domain (t) with boundaries moving with constant velocities.
FIG. 1.
- Centers of cells in (t old ) are shown with solid circles, and centers of cells in (tnew) -(t old ) are shown with unfilled circles.
- The authors solve (19) numerically on a rectangular domain with three elliptically-shaped holes, with boundary conditions computed using the exact solution (18).
- In the moving-boundary problem, the holes move with constant velocities.
- With both fixed and moving boundaries, the authors solve two separate problems with different boundary conditions: Dirichlet conditions on all boundaries; Dirichlet conditions on the fixed external boundaries, but Neumann conditions on the boundaries of the ellipses.
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Figures (15)
TABLE 3 
FIG. 6. Contour plots of absolute value of solution error to (19) at t = 1 for fixed boundaries, Dirichlet boundary conditions, h = 0:0125. Top figure is for Crank–Nicolson method, bottom figure for TGA method. 
FIG. 4. Solution error at t = 1 using Crank–Nicolson (stars) and TGA (circles), for the Dirichlet problem for (19) with fixed boundaries. Left-hand plot shows max norm, right-hand plot shows 1-norm. We see that both jj jj1 = O(h2) and jj jj1 = O(h2), indicating second-order accuracy. 
FIG. 5. Solution error at t = 1 using Crank–Nicolson (stars) and TGA (circles), for the Neumann problem for (19) with fixed boundaries. Left-hand plot shows max norm, right-hand plot shows 1-norm. We see that both jj jj1 = O(h2) and jj jj1 = O(h2), indicating second-order accuracy. 
FIG. 8. Contour plot of exact solution to (19) at t = 1 for moving-boundaryproblem. The dashed ellipses indicate the boundaries at t = 0. 
TABLE 1 
TABLE 2 Parameters for ellipses in (20), where = p 2=15 = 0:09428. 
FIG. 3. Contour plot of exact solution (18) to (19) at t = 1 with fixed boundaries. 
FIG. 10. Solution error at t = 1 in TGA method (circles) and Crank–Nicolson (stars), with Neumann conditions for (19) on moving boundaries. Left-hand plot shows max norm, right-hand plot shows 1-norm. 
FIG. 9. Solution error at t = 1 in TGA method (circles) and Crank–Nicolson method (stars), with Dirichlet conditions for (19) on moving boundaries. Left-hand plot shows max norm, right-hand plot shows 1-norm. 
FIG. 1. Centers of cells in (told) are shown with solid circles, and centers of cells in (tnew) - (told) are shown with unfilled circles. To estimate the value of U n at one of these latter points, we interpolate quadratically from values at the centers of three other cells in (told) forming a line with the new cell center. We pick whichever such line is closest in direction to the normal to the boundary at time t new. 
FIG. 2. From known values at points shown with solid circles on the moving boundary, we extrapolate to find values of (Dirichlet) or @ @n (Neumann) at points shown with unfilled circles, representing times told and tint on the boundary at time tnew. 
FIG. 11. Contour plot of absolute value of solution error to (19) at t = 1 for moving boundaries, Dirichlet boundary conditions, h = 0:0125 in TGA method. 
FIG. 12. Contour plot of absolute value of solution error to (19) at t = 1 for moving boundaries, Neumann boundary conditions, h = 0:0125 in TGA method. 
FIG. 7. Contour plots of absolute value of solution error to (19) at t = 1 for fixed boundaries, Neumann boundary conditions, h = 0:0125. Top figure is for Crank–Nicolson method, bottom figure for TGA method.
Citations
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TL;DR: In this paper, the authors present a methodology for simulating three-dimensional flow of incompressible viscoplastic fluids modeled by generalized Newtonian rheological equations, which is implemented in a highly efficient framework for massively parallelizable computations on blockstructured grids.
Abstract: We present a methodology for simulating three-dimensional flow of incompressible viscoplastic fluids modeled by generalized Newtonian rheological equations. It is implemented in a highly efficient framework for massively parallelizable computations on block-structured grids. In this context, geometric features are handled by the embedded boundary approach, which requires specialized treatment only in cells intersecting or adjacent to the boundary. This constitutes the first published implementation of an embedded boundary algorithm for simulating flow of viscoplastic fluids on structured grids. The underlying algorithm employs a two-stage Runge-Kutta method for temporal discretization, in which viscous terms are treated semi-implicitly and projection methods are utilized to enforce the incompressibility constraint. We augment the embedded boundary algorithm to deal with the variable apparent viscosity of the fluids. Since the viscosity depends strongly on the strain rate tensor, special care has been taken to approximate the components of the velocity gradients robustly near boundary cells, both for viscous wall fluxes in cut cells and for updates of apparent viscosity in cells adjacent to them. After performing convergence analysis and validating the code against standard test cases, we present the first ever fully three-dimensional simulations of creeping flow of Bingham plastics around translating objects. Our results shed new light on the flow fields around these objects.
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Cites methods from "A Cartesian grid embedded boundary ..."
...It has since been used to solve the heat equation [24, 29] and the incompressible flow on a time-dependent domain [25] with second order accuracy....
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...There are many different approach available as in [39, 14, 24]....
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...For an interior cell or a boundary cell, only one unknown is needed at the cell center and a standard finite volume method can be used to setup one algebraic equation using the elliptic equation as in [39, 14, 24, 29]....
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...2 Additive Runge-Kutta Method The implicit Runge-Kutta method ([36]) was used to solve the time dependent parabolic initial boundary value problem in ([24, 29]) and the parabolic interface problem in [39]....
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Dissertation•
26 Sep 2006
TL;DR: In this paper, Dirichlet, Robin ou Neumann et al. propose a schema numerique generique to traiter toutes les conditions aux limites generales.
Abstract: Ce travail est dedie a la mise en place de deux methodes originales de type domaine fictif pour la resolution de problemes elliptiques (de type convection-diffusion) avec des conditions aux limites generales et eventuellement mixtes : Dirichlet, Robin ou Neumann. L'originalite de ces methodes consiste a utiliser le maillage du domaine fictif, generalement non adapte a la geometrie du domaine physique, pour definir une frontiere immergee approchee sur laquelle seront appliquees les conditions aux limites immergees. Un meme schema numerique generique permet de traiter toutes les conditions aux limites generales. Ainsi, contrairement aux approches classiques de domaine fictif, ces methodes ne necessitent ni l'introduction d'un maillage surfacique de la frontiere immergee ni la modification locale du schema numerique. Deux modelisations de la frontiere immergee sont etudiees. Dans la premiere modelisation, appelee interface diffuse, la frontiere immergee approchee est l'union des mailles traversees par la frontiere originelle. Dans la deuxieme modelisation, la frontiere immergee est approchee par une interface dite fine s'appuyant sur les faces de cellules du maillage. Des conditions de transmissions algebriques combinant les sauts de la solution et du flux sont introduites sur cette interface fine. Pour ces deux modelisations, le probleme fictif a resoudre ainsi que le traitement des conditions aux limites immergees sont detailles. Un schema aux elements finis Q1 est utilise pour valider numeriquement le modele a interface diffuse alors qu'un nouveau schema aux volumes finis est developpe pour le modele a interface fine et sauts immerges. Chaque methode est combinee avec un algorithme de raffinement de maillage multi-niveaux (avec residu de solution ou du flux) autour de la frontiere immergee afin d'ameliorer la precision de la solution obtenue. Parallelement, une analyse theorique de convergence en maillage non adapte au domaine physique a ete effectuee pour une methode d'elements finis Q1. Cette etude demontre l'ordre de convergence des methodes de domaine fictif mises en place. Parmi les nombreuses applications industrielles possibles, une simulation sur une maquette d'echangeur de chaleur dans les centrales nucleaires permet d'apprecier la performance des methodes mises en oeuvre.
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Cites background from "A Cartesian grid embedded boundary ..."
...M) [Leveque et Li 1994, Li 2003], n domaine tronqué (C.G.E.M.) [Johansen et Colella 1998, McCorquodale et al. 2001]. u approche interface «diffuse» : n frontière en “anneau” [Rukhovets 1967], n méthode de frontière immergée (I.B....
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TL;DR: In this paper, a second-order-accurate finite-volume method is developed for the solution of incompressible NavierStokes equations on locally refined nested Cartesian grids.
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TL;DR: This method is second order accurate uniformly up to boundaries and is able to resolve small features without the use of fine meshes, necessary to enable the repeated simulations required for future verification and validation and uncertainty quantification studies of defibrillation.
Abstract: We present a sharp boundary electrocardiac simulation model based on the finite volume embedded boundary method for the solution of voltage dynamics in irregular domains with anisotropy and a high degree of anatomical detail. This method is second order accurate uniformly up to boundaries and is able to resolve small features without the use of fine meshes. This capability is necessary to enable the repeated simulations required for future verification and validation and uncertainty quantification studies of defibrillation, where fine-scale heterogeneities, such as those formed by small blood vessels, play an important role and require resolution.
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Cites background or methods from "A Cartesian grid embedded boundary ..."
...The treatment of the EBM follows ideas of [41, 61]....
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...The result, as analyzed in [41], is a second order accurate method in the L∞ norm, meaning that anomalous boundary signals will not occur in the method....
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...Originally proposed by Colella and others [32, 41, 54], the EBM method maintains sharp boundaries and interfaces in geometrically complex domains [52, 61]....
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References
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40,330 citations
Additional excerpts
...Similar approaches based on formally inconsistent discretizations at the irregular boundary have been used previously and observed to be stable [1, 9], so we expect that the extension to the more accurate boundary discretization should be straightforward....
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Book•
07 Jan 2013
TL;DR: In this article, Leray-Schauder and Harnack this article considered the Dirichlet Problem for Poisson's Equation and showed that it is a special case of Divergence Form Operators.
Abstract: Chapter 1. Introduction Part I: Linear Equations Chapter 2. Laplace's Equation 2.1 The Mean Value Inequalities 2.2 Maximum and Minimum Principle 2.3 The Harnack Inequality 2.4 Green's Representation 2.5 The Poisson Integral 2.6 Convergence Theorems 2.7 Interior Estimates of Derivatives 2.8 The Dirichlet Problem the Method of Subharmonic Functions 2.9 Capacity Problems Chapter 3. The Classical Maximum Principle 3.1 The Weak Maximum Principle 3.2 The Strong Maximum Principle 3.3 Apriori Bounds 3.4 Gradient Estimates for Poisson's Equation 3.5 A Harnack Inequality 3.6 Operators in Divergence Form Notes Problems Chapter 4. Poisson's Equation and Newtonian Potential 4.1 Holder Continuity 4.2 The Dirichlet Problem for Poisson's Equation 4.3 Holder Estimates for the Second Derivatives 4.4 Estimates at the Boundary 4.5 Holder Estimates for the First Derivatives Notes Problems Chapter 5. Banach and Hilbert Spaces 5.1 The Contraction Mapping 5.2 The Method of Cintinuity 5.3 The Fredholm Alternative 5.4 Dual Spaces and Adjoints 5.5 Hilbert Spaces 5.6 The Projection Theorem 5.7 The Riesz Representation Theorem 5.8 The Lax-Milgram Theorem 5.9 The Fredholm Alternative in Hilbert Spaces 5.10 Weak Compactness Notes Problems Chapter 6. Classical Solutions the Schauder Approach 6.1 The Schauder Interior Estimates 6.2 Boundary and Global Estimates 6.3 The Dirichlet Problem 6.4 Interior and Boundary Regularity 6.5 An Alternative Approach 6.6 Non-Uniformly Elliptic Equations 6.7 Other Boundary Conditions the Obliue Derivative Problem 6.8 Appendix 1: Interpolation Inequalities 6.9 Appendix 2: Extension Lemmas Notes Problems Chapter 7. Sobolev Spaces 7.1 L^p spaces 7.2 Regularization and Approximation by Smooth Functions 7.3 Weak Derivatives 7.4 The Chain Rule 7.5 The W^(k,p) Spaces 7.6 DensityTheorems 7.7 Imbedding Theorems 7.8 Potential Estimates and Imbedding Theorems 7.9 The Morrey and John-Nirenberg Estimes 7.10 Compactness Results 7.11 Difference Quotients 7.12 Extension and Interpolation Notes Problems Chapter 8 Generalized Solutions and Regularity 8.1 The Weak Maximum Principle 8.2 Solvability of the Dirichlet Problem 8.3 Diferentiability of Weak Solutions 8.4 Global Regularity 8.5 Global Boundedness of Weak Solutions 8.6 Local Properties of Weak Solutions 8.7 The Strong Maximum Principle 8.8 The Harnack Inequality 8.9 Holder Continuity 8.10 Local Estimates at the Boundary 8.11 Holder Estimates for the First Derivatives 8.12 The Eigenvalue Problem Notes Problems Chapter 9. Strong Solutions 9.1 Maximum Princiles for Strong Solutions 9.2 L^p Estimates: Preliminary Analysis 9.3 The Marcinkiewicz Interpolation Theorem 9.4 The Calderon-Zygmund Inequality 9.5 L^p Estimates 9.6 The Dirichlet Problem 9.7 A Local Maximum Principle 9.8 Holder and Harnack Estimates 9.9 Local Estimates at the Boundary Notes Problems Part II: Quasilinear Equations Chapter 10. Maximum and Comparison Principles 10.1 The Comparison Principle 10.2 Maximum Principles 10.3 A Counterexample 10.4 Comparison Principles for Divergence Form Operators 10.5 Maximum Principles for Divergence Form Operators Notes Problems Chapter 11. Topological Fixed Point Theorems and Their Application 11.1 The Schauder Fixes Point Theorem 11.2 The Leray-Schauder Theorem: a Special Case 11.3 An Application 11.4 The Leray-Schauder Fixed Point Theorem 11.5 Variational Problems Notes Chapter 12. Equations in Two Variables 12.1 Quasiconformal Mappings 12.2 holder Gradient Estimates for Linear Equations 12.3 The Dirichlet Problem for Uniformly Elliptic Equations 12.4 Non-Uniformly Elliptic Equations Notes Problems Chapter 13. Holder Estimates for
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01 Jan 1997
TL;DR: In this paper, a class of partial differential equations that generalize and are represented by Laplace's equation was studied. And the authors used the notation D i u, D ij u for partial derivatives with respect to x i and x i, x j and the summation convention on repeated indices.
Abstract: We study in this chapter a class of partial differential equations that generalize and are to a large extent represented by Laplace’s equation. These are the elliptic partial differential equations of second order. A linear partial differential operator L defined by
$$ Lu{\text{: = }}{a_{ij}}\left( x \right){D_{ij}}u + {b_i}\left( x \right){D_i}u + c\left( x \right)u $$
is elliptic on Ω ⊂ ℝ n if the symmetric matrix [a ij ] is positive definite for each x ∈ Ω. We have used the notation D i u, D ij u for partial derivatives with respect to x i and x i , x j and the summation convention on repeated indices is used. A nonlinear operator Q,
$$ Q\left( u \right): = {a_{ij}}\left( {x,u,Du} \right){D_{ij}}u + b\left( {x,u,Du} \right) $$
[D u = (D 1 u, ..., D n u)], is elliptic on a subset of ℝ n × ℝ × ℝ n ] if [a ij (x, u, p)] is positive definite for all (x, u, p) in this set. Operators of this form are called quasilinear. In all of our examples the domain of the coefficients of the operator Q will be Ω × ℝ × ℝ n for Ω a domain in ℝ n . The function u will be in C 2(Ω) unless explicitly stated otherwise.
8,299 citations
"A Cartesian grid embedded boundary ..." refers background in this paper
...However, it is well known that, for any domain with smooth boundary, a smooth function can be extended to all of R with a bound on the relative increase in the C norms that depends only on the domain and (k; ) [5]....
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TL;DR: In this paper, a second-order projection method for the Navier-Stokes equations is proposed, which uses a specialized higher-order Godunov method for differencing the nonlinear convective terms.
1,287 citations
"A Cartesian grid embedded boundary ..." refers methods in this paper
...The method described here, together with that in [6] for elliptic PDEs and [8] for hyperbolic PDEs, provide the fundamental components required for developing second-order accurate methods for a broad range of continuum mechanics problems in irregular geometries based on the predictor–corrector approach in [2]....
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TL;DR: A numerical method for solving Poisson's equation, with variable coefficients and Dirichlet boundary conditions, on two-dimensional regions using a finite-volume discretization, which embeds the domain in a regular Cartesian grid.
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"A Cartesian grid embedded boundary ..." refers background or methods in this paper
...As in previous work on elliptic problems [6], our approach uses a finite-volume discretization, which embeds the domain in a regular Cartesian grid....
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...We follow the approach described in [6, 7]....
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...With Dirichlet boundary conditions as from (3), we compute an estimate of ∂ψ ∂n by interpolating from the grid values and the values at the boundaries; for details, see [6]....
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...The method described here, together with that in [6] for elliptic PDEs and [8] for hyperbolic PDEs, provide the fundamental components required for developing second-order accurate methods for a broad range of continuum mechanics problems in irregular geometries based on the predictor–corrector approach in [2]....
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...This is routine for the case in which the embedded boundary is contained in the finest level of refinement [6], but requires some additional discretization design when the embedded boundary crosses coarse–fine interfaces....
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