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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09 Jan 2006
TL;DR: A new high-order immersed interface method for elliptic equations with imbedded interface of discontinuity that can be arbitrarily high- order accurate, and easily applied to practical two-phase flow because only the physical jump conditions for variables and their first derivatives are needed in the finite difference formulas.
Abstract: Numerical simulation of two-phase flow with imbedded interface of discontinuity is challenging in two major aspects. First, the interface can undergo change, merge and breakup during the course of the simulation. Examples of current successful methods in modeling flow with interface are, among others, the volume-of-fluid method, fronttracking method, and level-set method. Second, flow variables and their derivatives can be discontinuous across the interface. This discontinuity poses severe limitation on the accuracy of commonly used numerical methods. Current available methods in treating the interface discontinuity, such as the immersed boundary method and the ghost fluid method, are mostly first order accurate. Though the immersed interface method of LeVeque and Li (1994) can be globally second order, it is often difficult to apply to complex multi-dimensional flow. This paper presents a new high-order immersed interface method for elliptic equations with imbedded interface of discontinuity. The new method can be arbitrarily high-order accurate, and it can be easily applied to practical two-phase flow because only the physical jump conditions for variables and their first derivatives are needed in the finite difference formulas. In addition, the new interface difference formulas are expressed in a general explicit form so that they can be applied to different multi-dimensional problems without any modification. The new interface algorithms of up to accuracy have been tested for one and two-dimensional elliptic equations with imbedded interface. The extension to practical two-phase flow applications will be presented in a future paper.
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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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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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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.
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"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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"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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