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
A new high-order immersed interface method for solving elliptic equations with imbedded interface of discontinuity
TL;DR: Test results show that the new fourth-order immersed interface method leads to a significant improvement in accuracy of the numerical solutions and has potential advantages in the application to two-phase flow because of its high-order accuracy and simplicity in applications.
Journal ArticleDOI
A high-order cut-cell method for numerical simulation of hypersonic boundary-layer instability with surface roughness
TL;DR: A new high-order cut-cell method is presented to overcome the natural complexities in grid generation around arbitrary surface of roughness in hypersonic boundary layers.
Journal ArticleDOI
A general fictitious domain method with immersed jumps and multilevel nested structured meshes
TL;DR: The numerical results reported here for convection-diffusion problems with Dirichlet, Robin and mixed (Dirichlet and Robin) boundary conditions confirm the expected accuracy as well as the performances of the present method.
Book ChapterDOI
A generative approach for image-based modeling of tumor growth
Bjoern H. Menze,Koen Van Leemput,Antti Honkela,Ender Konukoglu,Marc-André Weber,Nicholas Ayache,Polina Golland +6 more
TL;DR: In this paper, a joint generative model of tumor growth and image observation is proposed for analyzing imaging data in patients with glioma, which can be used for integrating information from different multi-modal imaging protocols.
Journal Article
A Cartesian grid embedded boundary method for solving the Poisson and heat equations with discontinuous coefficients in three dimensions
TL;DR: In this paper, a Cartesian cut-cell/embedded boundary method is used to represent the interface between materi- als, as described in Johansen & Colella (1998).
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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