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.
Abstract: We present an algorithm for solving the heat equation on irregular time-dependent domains. It is based on the Cartesian grid embedded boundary algorithm of Johansen and Colella (J. Comput. Phys. 147(2):60--85) for discretizing Poisson's equation, combined with a second-order accurate discretization of the time derivative. This leads to a method that is second-order accurate in space and time. For the case where the boundary is moving, we convert the moving-boundary problem to a sequence of fixed-boundary problems, combined with an extrapolation procedure to initialize values that are uncovered as the boundary moves. We find that, in the moving boundary case, the use of Crank--Nicolson time discretization is unstable, requiring us to use the L{sub 0}-stable implicit Runge--Kutta method of Twizell, Gumel, and Arigu.
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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01 Nov 2002
TL;DR: An efficient ghost-cell immersed boundary method (GCIBM) for simulating turbulent flows in complex geometries is presented in this paper, where a boundary condition is enforced through a ghost cell method.
Abstract: An efficient ghost-cell immersed boundary method (GCIBM) for simulating turbulent flows in complex geometries is presented. A boundary condition is enforced through a ghost cell method. The reconstruction procedure allows systematic development of numerical schemes for treating the immersed boundary while preserving the overall second-order accuracy of the base solver. Both Dirichlet and Neumann boundary conditions can be treated. The current ghost cell treatment is both suitable for staggered and non-staggered Cartesian grids. The accuracy of the current method is validated using flow past a circular cylinder and large eddy simulation of turbulent flow over a wavy surface. Numerical results are compared with experimental data and boundary-fitted grid results. The method is further extended to an existing ocean model (MITGCM) to simulate geophysical flow over a three-dimensional bump. The method is easily implemented as evidenced by our use of several existing codes.
740 citations
Cites background from "A Cartesian grid embedded boundary ..."
...[31] extended this approach to the solution of the time-dependent heat equation....
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TL;DR: An efficient ghost-cell immersed boundary method (GCIBM) for simulating turbulent flows in complex geometries is presented in this article, where a boundary condition is enforced through a ghost cell method.
Abstract: An efficient ghost-cell immersed boundary method (GCIBM) for simulating turbulent flows in complex geometries is presented. A boundary condition is enforced through a ghost cell method. The reconstruction procedure allows systematic development of numerical schemes for treating the immersed boundary while preserving the overall second-order accuracy of the base solver. Both Dirichlet and Neumann boundary conditions can be treated. The current ghost cell treatment is both suitable for staggered and non-staggered Cartesian grids. The accuracy of the current method is validated using flow past a circular cylinder and large eddy simulation of turbulent flow over a wavy surface. Numerical results are compared with experimental data and boundary-fitted grid results. The method is further extended to an existing ocean model (MITGCM) to simulate geophysical flow over a three-dimensional bump. The method is easily implemented as evidenced by our use of several existing codes.
674 citations
TL;DR: The results obtaining by the adaptive method show good qualitative agreement with simulation results obtained by earlier non-adaptive versions of the method, but the flow in the vicinity of the model heart valves indicates that the new methodology provides enhanced boundary layer resolution.
Abstract: Like many problems in biofluid mechanics, cardiac mechanics can be modeled as the dynamic interaction of a viscous incompressible fluid (the blood) and a (visco-)elastic structure (the muscular walls and the valves of the heart). The immersed boundary method is a mathematical formulation and numerical approach to such problems that was originally introduced to study blood flow through heart valves, and extensions of this work have yielded a three-dimensional model of the heart and great vessels. In the present work, we introduce a new adaptive version of the immersed boundary method. This adaptive scheme employs the same hierarchical structured grid approach (but a different numerical scheme) as the two-dimensional adaptive immersed boundary method of Roma et al. [A multilevel self adaptive version of the immersed boundary method, Ph.D. Thesis, Courant Institute of Mathematical Sciences, New York University, 1996; An adaptive version of the immersed boundary method, J. Comput. Phys. 153 (2) (1999) 509–534] and is based on a formally second order accurate (i.e., second order accurate for problems with sufficiently smooth solutions) version of the immersed boundary method that we have recently described [B.E. Griffith, C.S. Peskin, On the order of accuracy of the immersed boundary method: higher order convergence rates for sufficiently smooth problems, J. Comput. Phys. 208 (1) (2005) 75–105]. Actual second order convergence rates are obtained for both the uniform and adaptive methods by considering the interaction of a viscous incompressible flow and an anisotropic incompressible viscoelastic shell. We also present initial results from the application of this methodology to the three-dimensional simulation of blood flow in the heart and great vessels. The results obtained by the adaptive method show good qualitative agreement with simulation results obtained by earlier non-adaptive versions of the method, but the flow in the vicinity of the model heart valves indicates that the new methodology provides enhanced boundary layer resolution. Differences are also observed in the flow about the mitral valve leaflets.
356 citations
Cites methods from "A Cartesian grid embedded boundary ..."
...Although both adaptive schemes employ projection methods to solve the incompressible Navier–Stokes equations, the present work employs a cell-centered projection method that makes use of an implicit L-stable discretization of the viscous terms [15,16] and a second order Godunov method for the explicit treatment of the nonlinear advection terms [17–20]....
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...[16] is used to integrate the viscous terms in time....
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TL;DR: In this article, Lai et al. describe a new formally second order accurate immersed boundary method and demonstrate its performance for a prototypical fluid-structure interaction problem, involving an immersed viscoelastic shell of finite thickness, studied over a broad range of Reynolds numbers.
Abstract: The immersed boundary method is both a mathematical formulation and a numerical scheme for problems involving the interaction of a viscous incompressible fluid and a (visco-)elastic structure. In [M.-C. Lai, Simulations of the flow past an array of circular cylinders as a test of the immersed boundary method, Ph.D. thesis, Courant Institute of Mathematical Sciences, New York University, 1998; M.-C. Lai, C.S. Peskin, An immersed boundary method with formal second-order accuracy and reduced numerical viscosity, J. Comput. Phys. 160 (2000) 705-719], Lai and Peskin introduced a formally second order accurate immersed boundary method, but the convergence properties of their algorithm have only been examined computationally for problems with nonsmooth solutions. Consequently, in practice only first order convergence rates have been observed. In the present work, we describe a new formally second order accurate immersed boundary method and demonstrate its performance for a prototypical fluid-structure interaction problem, involving an immersed viscoelastic shell of finite thickness, studied over a broad range of Reynolds numbers. We consider two sets of material properties for the viscoelastic structure, including a case where the material properties of the coupled system are discontinuous at the fluid-structure interface. For both sets of material properties, the true solutions appear to possess sufficient smoothness for the method to converge at a second order rate for fully resolved computations.
299 citations
Cites methods from "A Cartesian grid embedded boundary ..."
...projection method that makes use of an implicit L-stable discretization of the viscous terms [15,16] and a second order Godunov method for the explicit treatment of the nonlinear advection terms [17–19]....
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...[16] is used to integrate the viscous terms in time....
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TL;DR: A general approach for solving partial differential equations in complex, stationary, or moving geometries with Dirichlet, Neumann, and Robin boundary conditions with matched asymptotic expansions is presented.
Abstract: We extend previous work and present a general approach for solving partial differential equations in complex, stationary, or moving geometries with Dirichlet, Neumann, and Robin boundary conditions. Using an implicit representation of the geometry through an auxilliary phase field function, which replaces the sharp boundary of the domain with a diffuse layer (e.g. diffuse domain), the equation is reformulated on a larger regular domain. The resulting partial differential equation is of the same order as the original equation, with additional lower order terms to approximate the boundary conditions. The reformulated equation can be solved by standard numerical techniques. We use the method of matched asymptotic expansions to show that solutions of the re-formulated equations converge to those of the original equations. We provide numerical simulations which confirm this analysis. We also present applications of the method to growing domains and complex three-dimensional structures and we discuss applications to cell biology and heteroepitaxy.
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Additional excerpts
..., [20, 21, 22, 32, 33, 43]), the immersed interface method (e....
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References
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40,330 citations
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...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
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]....
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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.
Abstract: In this paper we describe a second-order projection method for the time-dependent, incom pressible Navier-Stokes equations. As in the original projection method developed by Chorin, we first solve diffusion-convection equations to predict intermediate velocities which are then projected onto the space of divergence-free vector fields. By introducing more coupling between the diffusion--{;onvection step and the projection step we obtain a temporal discretiza tion that is second-order accurate. Our treatment of the diffusion-convection step uses a specialized higher order Godunov method for differencing the nonlinear convective terms that provides a robust treatment of these terms at high Reynolds number. The Godunov procedure is second-order accurate for smooth flow and remains stable for discontinuous initial data, even in the zero-viscosity limit. We approximate the projection directly using a Galerkin procedure that uses a local basis for discretely divergence-free vector fields. Numerical results are presented validating the convergence properties of the method. We also apply the method to doubly periodic shear-layers to assess the performance of the method on more difficult
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.
Abstract: We present a numerical method for solving Poisson's equation, with variable coefficients and Dirichlet boundary conditions, on two-dimensional regions. The approach uses a finite-volume discretization, which embeds the domain in a regular Cartesian grid. We treat the solution as a cell-centered quantity, even when those centers are outside the domain. Cells that contain a portion of the domain boundary use conservative differencing of second-order accurate fluxes on each cell volume. The calculation of the boundary flux ensures that the conditioning of the matrix is relatively unaffected by small cell volumes. This allows us to use multigrid iterations with a simple point relaxation strategy. We have combined this with an adaptive mesh refinement (AMR) procedure. We provide evidence that the algorithm is second-order accurate on various exact solutions and compare the adaptive and nonadaptive calculations.
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....
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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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