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
TLDR
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).read more
Figures
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
Citations
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Journal ArticleDOI
High-Resolution Simulation of Pore-Scale Reactive Transport Processes Associated with Carbon Sequestration
TL;DR: The authors present a direct numerical simulation modeling capability that can resolve flow and transport processes in geometric features obtained from the image data of realistic pore space at unprecedented scale and resolution and demonstrate the scalability of this new capability, known as Chombo-Crunch.
Journal ArticleDOI
Flow simulations in arbitrarily complex cardiovascular anatomies – An unstructured Cartesian grid approach
Diane de Zélicourt,Liang Ge,Chang Wang,Fotis Sotiropoulos,Anvar Gilmanov,Anvar Gilmanov,Ajit P. Yoganathan +6 more
TL;DR: The challenging clinical scenario of a single-ventricle patient with severe arterio-venous malformations is tackled, seeking to provide a fluid dynamics prospective on a clinical problem and suggestions for procedure improvements.
Journal ArticleDOI
Vessel Segmentation and Blood Flow Simulation Using Level-Sets and Embedded Boundary Methods
TL;DR: This method directly simulates the blood flow inside the extracted surface without losing any complicated details and without building additional grids.
Journal ArticleDOI
Effects of geometry discretization aspects on the numerical solution of the bioheat transfer equation with the FDTD technique.
TL;DR: Two issues that have to be taken into consideration for accurate thermal modelling with the finite-difference time-domain (FDTD) method, namely the tissue interfaces and the staircasing effect are highlighted.
Journal ArticleDOI
The moving boundary node method: A level set-based, finite volume algorithm with applications to cell motility
Charles W. Wolgemuth,Mark Zajac +1 more
TL;DR: An algorithm is developed that uses the level set method to move the cell boundary and uses information stored in the distance map to construct a finite volume representation of the cell, which preserves Cartesian connectivity of nodes in the finiteVolume representation while resolving the distorted cell geometry.
References
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Book
Elliptic Partial Differential Equations of Second Order
David Gilbarg,Neil S. Trudinger +1 more
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.
Book ChapterDOI
Elliptic Partial Differential Equations of Second Order
Piero Bassanini,Alan R. Elcrat +1 more
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.
Journal ArticleDOI
A second-order projection method for the incompressible navier-stokes equations
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.
Journal ArticleDOI
A Cartesian Grid Embedded Boundary Method for Poisson's Equation on Irregular Domains
Hans Johansen,Phillip Colella +1 more
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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