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
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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
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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
A simple weak formulation for solving two-dimensional diffusion equation with local reaction on the interface
TL;DR: Numerical experiments show that this method for solving two-dimensional diffusion equation with nonhomogeneous jump condition and nonlinear flux jump condition located at the interface is nearly second order accurate in the L ∞ norm.
DissertationDOI
A parallel second order Cartesian method for elliptic interface problems and its application to tumor growth model
TL;DR: A parallel Cartesian method to solve elliptic problems with complex interfaces and its application to elliptic irregular domain problems in the framework of a tumor growth model, the two-species Darcy model is applied.
Dissertation
Development and use of interface-capturing methods for investigation of surfactant-covered drops in electric fields
TL;DR: In this article, the authors investigated the development and use of interface-capturing methods for detailed simulations of surfactant-covered drops in electric fields, and established a mathematical model for the simulation.
Proceedings ArticleDOI
Stable, High-Order and Conservative Cut-Cell Methods
Peter Brady,Daniel Livescu +1 more
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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