Convergence analysis of the finite difference ADI scheme for the heat equation on a convex set

2021 ◽  
pp. 1
Author(s):  
Bernard Bialecki ◽  
Maxsymillian Dryja ◽  
Ryan I. Fernandes
Keyword(s):  
Finite Difference ◽  
Heat Equation ◽  
Convergence Analysis ◽  
Convex Set ◽  
Adi Scheme

scholarly journals The convergence of a numerical method for total variation flow

2021 ◽  
Vol 15 ◽  
pp. 174830262110113
Author(s):  
Qianying Hong ◽  
Ming-jun Lai ◽  
Jingyue Wang
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Numerical Method ◽  
Convergence Analysis ◽  
Total Variation ◽  
Iterative Algorithm ◽  
Finite Difference Scheme ◽  
Gradient Flow ◽  
Time Dependent ◽  
Total Variation Flow

We present a convergence analysis for a finite difference scheme for the time dependent partial different equation called gradient flow associated with the Rudin-Osher-Fetami model. We devise an iterative algorithm to compute the solution of the finite difference scheme and prove the convergence of the iterative algorithm. Finally computational experiments are shown to demonstrate the convergence of the finite difference scheme.

Optimized Icosahedral Grids: Performance of Finite-Difference Operators and Multigrid Solver

2013 ◽  
Vol 141 (12) ◽  
pp. 4450-4469 ◽  
Author(s):  
Ross P. Heikes ◽  
David A. Randall ◽  
Celal S. Konor
Keyword(s):  
Finite Difference ◽  
Convergence Analysis ◽  
Optimization Methods ◽  
Difference Operators ◽  
Grid Spacing ◽  
Multigrid Solver ◽  
Poisson Equations ◽  
The Earth ◽  
Grid Optimization ◽  
Accuracy Performance

Abstract This paper discusses the generation of icosahedral hexagonal–pentagonal grids, optimization of the grids, how optimization affects the accuracy of finite-difference Laplacian, Jacobian, and divergence operators, and a parallel multigrid solver that can be used to solve Poisson equations on the grids. Three different grid optimization methods are compared through an error convergence analysis. The optimization process increases the accuracy of the operators. Optimized grids up to 1-km grid spacing over the earth have been created. The accuracy, performance, and scalability of the multigrid solver are demonstrated.

Numerical identification of temperature dependent thermal conductivity using least squares method

2019 ◽  
Vol 30 (6) ◽  
pp. 3083-3099
Author(s):  
Anna Ivanova ◽  
Stanislaw Migorski ◽  
Rafal Wyczolkowski ◽  
Dmitry Ivanov
Keyword(s):  
Thermal Conductivity ◽  
Finite Difference ◽  
Heat Equation ◽  
Difference Scheme ◽  
Least Squares ◽  
Finite Difference Scheme ◽  
Nonlinear Heat Equation ◽  
Temperature Dependent ◽  
Content Type ◽  
Temperature Dependent Thermal Conductivity

Purpose This paper aims to considered the problem of identification of temperature-dependent thermal conductivity in the nonstationary, nonlinear heat equation. To describe the heat transfer in the furnace charge occupied by a homogeneous porous material, the heat equation is formulated. The inverse problem consists in finding the heat conductivity parameter, which depends on the temperature, from the measurements of the temperature in fixed points of the material. Design/methodology/approach A numerical method based on the finite-difference scheme and the least squares approach for numerical solution of the direct and inverse problems has been recently developed. Findings The influence of different numerical scheme parameters on the accuracy of the identified conductivity coefficient is studied. The results of the experiment carried out on real measurements are presented. Their results confirm the ones obtained earlier by using other methods. Originality/value Novelty is in a new, easy way to identify thermal conductivity by known temperature measurements. This method is based on special finite-difference scheme, which gives a resolvable system of algebraic equations. The results sensitivity on changes in the method parameters was studies. The algorithms of identification in the case of a purely mathematical experiment and in the case of real measurements, their differences and the practical details are presented.

On Finite-Difference Solutions of the Heat Equation in Spherical Coordinates

1987 ◽  
Vol 12 (4) ◽  
pp. 457-474 ◽  
Author(s):  
Jules Thibault ◽  
Simon Bergeron ◽  
Hugues Bonin
Keyword(s):  
Finite Difference ◽  
Heat Equation ◽  
Spherical Coordinates
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