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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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
Numerical computation of diffusion on a surface
Peter Schwartz,David Adalsteinsson,Phillip Colella,Adam P. Arkin,Adam P. Arkin,Matthew Onsum +5 more
TL;DR: It is demonstrated that the method is second-order accurate in space and time and is capable of computing solutions on complex surface geometries obtained from image data of cells.
Journal ArticleDOI
A Cartesian grid embedded boundary method for solving the Poisson and heat equations with discontinuous coefficients in three dimensions
TL;DR: Overall second order accuracy is achieved, as indicated by an array of tests using non-trivial interface geometries, and the ability of the method to solve problems with realistic physical parameters is demonstrated.
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: A joint generative model of tumor growth and of image observation that naturally handles multimodal and longitudinal data is proposed that can be used for integrating information from different multi-modal imaging protocols and can be adapted to other tumor growth models.
Journal ArticleDOI
Spatially adaptive stochastic numerical methods for intrinsic fluctuations in reaction-diffusion systems
TL;DR: The discretization approaches introduced for SPDEs have the potential to be widely applicable in the development of numerical methods for the study of spatially extended stochastic systems.
Journal ArticleDOI
A kernel-free boundary integral method for elliptic boundary value problems
Wenjun Ying,Craig S. Henriquez +1 more
TL;DR: It is observed that the number of GM-RES iterations used by the KFBI method for solving isotropic and moderately anisotropic BVPs is independent of the sizes of the grids that are employed to approximate the boundary and volume integrals.
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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