A Four-Step Fixed-Grid Method for 1D Stefan Problems

2010 ◽  
Vol 132 (11) ◽  
Author(s):  
M. Tadi
Keyword(s):  
Free Boundary ◽  
Moving Boundary ◽  
Free Boundary Problems ◽  
Grid Method ◽  
Moving Boundary Problems ◽  
Boundary Problems ◽  
One Dimensional ◽  
Nonlinear Part ◽  
Stefan Problems ◽  
Fixed Grid

This note is concerned with a fixed-grid finite difference method for the solution of one-dimensional free boundary problems. The method solves for the field variables and the location of the boundary in separate steps. As a result of this decoupling, the nonlinear part of the algorithm involves only a scalar unknown, which is the location of the moving boundary. A number of examples are used to study the applicability of the method. The method is particularly useful for moving boundary problems with various conditions at the front.

scholarly journals A numerical scheme for moving boundary problems that are gradient flows for the area functional

2000 ◽  
Vol 11 (1) ◽  
pp. 61-80 ◽  
Author(s):  
UWE F. MAYER
Keyword(s):  
Free Boundary ◽  
Mean Curvature ◽  
Numerical Scheme ◽  
Moving Boundary ◽  
Numerical Solutions ◽  
Free Boundary Problems ◽  
Moving Boundary Problems ◽  
Gradient Flows ◽  
Boundary Problems ◽  
Area Functional

Many moving boundary problems that are driven in some way by the curvature of the free boundary are gradient flows for the area of the moving interface. Examples are the Mullins–Sekerka flow, the Hele-Shaw flow, flow by mean curvature, and flow by averaged mean curvature. The gradient flow structure suggests an implicit finite differences approach to compute numerical solutions. The proposed numerical scheme will allow us to treat such free boundary problems in both IR2 and IR3. The advantage of such an approach is the re-usability of much of the setup for all of the different problems. As an example of the method, we compute solutions to the averaged mean curvature flow that exhibit the formation of a singularity.

Solving moving boundary problems with an adaptive moving grid method

1998 ◽  
Vol 53 (19) ◽  
pp. 3393-3411 ◽  
Author(s):  
Jörg Frauhammer ◽  
Harald Klein ◽  
Gerhart Eigenberger ◽  
Ulrich Nowak
Keyword(s):  
Moving Boundary ◽  
Grid Method ◽  
Moving Boundary Problems ◽  
Moving Grid ◽  
Boundary Problems

Heat Potentials Method in the Treatment of One-dimensional Free Boundry Problems Applied in Cryomedicine

2018 ◽  
Vol 85 (1-2) ◽  
pp. 111 ◽  
Author(s):  
Fatimat K. Kudayeva ◽  
Arslan A. Kaigermazov ◽  
Elizaveta K. Edgulova ◽  
Mariya M. Tkhabisimova ◽  
Aminat R. Bechelova
Keyword(s):  
Free Boundary ◽  
Integral Equations ◽  
Free Boundary Problems ◽  
Evolutionary Equation ◽  
Fredholm Integral Equations ◽  
Integral Expression ◽  
Boundary Problems ◽  
System Of Integral Equations ◽  
One Dimensional ◽  
Heat Potential

Free boundary problems are considered to be the most difcult and the least researched in the eld of mathematical physics. The present article is concerned with the research of the following issue: treatment of one-dimensional free boundary problems. The treated problem contains a nonlinear evolutionary equation, which occurs within the context of mathematical modeling of cryosurgery problems. In the course of the research, an integral expression has been obtained. The obtained integral expression presents a general solution to the non-homogeneous evolutionary equation which contains the functions that represent simple-layer and double-layer heat potential density. In order to determine the free boundary and the density of potential a system of nonlinear, the second kind of Fredholm integral equations was obtained within the framework of the given work. The treated problem has been reduced to the system of integral equations. In order to reduce the problem to the integral equation system, a method of heat potentials has been used. In the obtained system of integral equations instead of K(ξ; x; τ - t) in case of Dirichlet or Neumann conditions the corresponding Greens functions G(ξ; x; τ - t) or N(ξ; x; τ - t) have been applied. Herewith the integral expression contains fewer densities, but the selection of arbitrary functions is reserved. The article contains a number of results in terms of building a mathematical model of cooling and freezing processes of biological tissue, as well as their effective solution development.

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