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
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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
High-order finite-volume methods on locally-structured grids
TL;DR: An approach to designing arbitrarily high-order finite-volume spatial discretizations on locally-rectangular grids is presented, based on the use of a simple class of high- order quadratures for computing the average of fluxes over faces.
Journal ArticleDOI
A high-order multi-zone cut-stencil method for numerical simulations of high-speed flows over complex geometries
TL;DR: The work presented in this dissertation was motivated by a desire to study the effects of isolated roughness elements on the stability of hypersonic boundary layers and a new code was developed which can perform high-order direct numerical simulations of high-speed flows over arbitrary geometries.
Journal ArticleDOI
A high-order numerical approach with Cartesian meshes for modeling of wave propagation and heat transfer on irregular domains with inhomogeneous materials
Alexander Idesman,Bikash Dey +1 more
TL;DR: In this article, the authors proposed a new numerical approach for PDEs with constant coefficients on irregular domains and Cartesian meshes, based on the representation of the stencil coefficients as functions of the mesh size.
Journal ArticleDOI
A Numerical Handling of the Boundary Conditions Imposed by the Skull on an Inhomogeneous Diffusion-Reaction Model of Glioblastoma Invasion Into the Brain: Clinical Validation Aspects:
TL;DR: A novel explicit triscale reaction-diffusion numerical model of glioblastoma multiforme tumor growth is presented and a theoretical exploration suggests that a rough but still quite informative value of the doubling time may be calculated based on a homogeneous brain model.
Proceedings ArticleDOI
Air-flow simulation in realistic models of the trachea
TL;DR: A new technique for flow simulation in realistic anatomical airways that directly simulates the air-flow inside the extracted surface without losing any complicated details and without building additional grids is presented.
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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