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.
Did you find this useful? Give us your feedback
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
More filters
TL;DR: In this article , an extended multiple-gradient descent (MGD) method is introduced during the training phase to adaptively balance the interplay between different terms in the loss function, and the effectiveness, accuracy and robustness of the proposed framework are demonstrated through a collection of interface problems in two and three spatial dimensions, including a moving interface case.
2 citations
01 Jan 2007
TL;DR: In this article, a projection method for the solution of diffusive transport and reaction equations of electrochemical systems on irregular time-dependent domains is presented, and the resulting method is first order accurate in time, and is observed to be stable for relatively large time steps.
Abstract: We present a projection method for the solution of the diffusive transport and reaction equations of electrochemical systems on irregular time-dependent domains. Specific applications include electrodeposition of copper in sub-micron trenches, as well as any other electrochemical system with an arbitrarily shaped bulk region of dilute electrolyte solution. Our method uses a finite volume spatial discretization that is second-order accurate throughout, including a nonuniform region used as a transition to the far-field chemical concentrations. Time integration is performed with a splitting technique that includes a projection step to solve for the electric potential. The resulting method is first order accurate in time, and is observed to be stable for relatively large time steps. Furthermore, the algorithm complexity scales very respectably with grid refinement and is naturally parallelizable.
2 citations
25 Sep 2008
TL;DR: A three-dimensional parallel electromagnetic code is developed to simulate high current electron devices with complex geometry, where particle-in-cell method is used.
Abstract: A three-dimensional parallel electromagnetic code is developed to simulate high current electron devices with complex geometry, where particle-in-cell method is used. For the complex geometry, we treat geometrically complex features embeds irregular geometry information into a Cartesian mesh. Also, a time integration algorithm is provided to solve the fully electromagnetic problem. A typical device with complex geometry is simulated by the parallel code on 128 cpus.
2 citations
Cites methods from "A Cartesian grid embedded boundary ..."
...An emerging methodology for treating geometrically complex features embeds irregular geometry information into a Cartesian mesh [2, 3]....
[...]
01 Jan 2012
TL;DR: A second-order accurate projection method to solve the incompressible Navier-Stokes equations on irregular domains in two and three dimensions using a finite-volume discretization obtained from intersecting the irregular domain boundary with a Cartesian grid is presented.
Abstract: We present a second-order accurate projection method to solve the incompressible Navier-Stokes equations on irregular domains in two and three dimensions. We use a finite-volume discretization obtained from intersecting the irregular domain boundary with a Cartesian grid. We address the small-cell stability problem associated with such methods by hybridizing a conservative discretization of the advective terms with a stable, nonconservative discretization at irregular control volumes, and redistributing the difference to nearby cells. Our projection is based upon a finite-volume discretization of Poisson’s equation. We use a second-order, L∞-stable algorithm to advance in time. Block structured local refinement is applied in space. The resulting method is second-order accurate in L1 for smooth problems. We demonstrate the method on benchmark problems for flow past a cylinder in 2D and a sphere in 3D as well as flows in 3D geometries obtained from image data.
2 citations
01 Jan 2012
TL;DR: A Stable Algorithm for Non-Negative Invariant Numerical Solution of Reaction-Diffusion Systems on Complicated Domains as discussed by the authors is a stable algorithm for non-negative and invariant numerical solution of reaction-diffusion systems on complicated domains.
Abstract: A Stable Algorithm for Non-Negative Invariant Numerical Solution of Reaction-Diffusion Systems on Complicated Domains
2 citations
References
More filters
[...]
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....
[...]
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
18,443 citations
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]....
[...]
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]....
[...]
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.
470 citations
"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....
[...]
...We follow the approach described in [6, 7]....
[...]
...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]....
[...]
...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]....
[...]
...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....
[...]