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Title: Solving diffusion equations with rough coefficients in rough grids

Abstract

A finite-difference algorithm for the numerical solution of diffusion problems in strongly heterogeneous and nonisotropic media is constructed for logically rectangular grids. The performance of this algorithm is comparable to other algorithms for problems with smooth coefficients and regular grids, and it is superior for problems with rough coefficients and/or skewed grids. The algorithm is derived using the support-operators method, which constructs discrete analogs of the divergence and flux operator that satisfy discrete analogs of the important integral identities relating the continuum operators. This paper gives the first application of this method to the solution of diffusion problems in heterogeneous an nonisotropic media. The support-operators method forces the discrete analog of the flux operator to be the negative adjoint of the discrete divergence in an inner product weighted by the conductivity, as in the differential case. Once this is accomplished, many other important properties follow; for example, the scheme is conservative and the discrete analog of the variable material Laplacian is symmetric and negative definite. In addition, on any grid, the discrete divergence is zero on constant vectors and the discrete divergence is zero on constant vectors and the discrete divergence is zero on constant vectors and the discrete fluxmore » operator is exact for linear functions in case when K is piecewise constant. Moreover, the discrete gradient`s null space is the constant functions, just as in the continuum. Because the algorithm is flux based, it has twice as many unknowns as more standard algorithms. However, the matrices that need to be inverted are symmetric and positive definite, so the most powerful linear solvers can be applied. Also, the scheme is second-order accurate so, all things considered, it is efficient.« less

Authors:
 [1];  [2]
  1. Los Alamos National Lab., NM (United States)
  2. Univ. of New Mexico, Albuquerque, NM (United States)
Publication Date:
OSTI Identifier:
530648
DOE Contract Number:  
W-7405-ENG-36
Resource Type:
Journal Article
Journal Name:
Journal of Computational Physics
Additional Journal Information:
Journal Volume: 129; Journal Issue: 2; Other Information: PBD: Dec 1996
Country of Publication:
United States
Language:
English
Subject:
66 PHYSICS; 99 MATHEMATICS, COMPUTERS, INFORMATION SCIENCE, MANAGEMENT, LAW, MISCELLANEOUS; DIFFUSION; PARTIAL DIFFERENTIAL EQUATIONS; FINITE DIFFERENCE METHOD; ALGORITHMS; NUMERICAL SOLUTION; MESH GENERATION; BOUNDARY CONDITIONS

Citation Formats

Shashkov, M, and Steinberg, S. Solving diffusion equations with rough coefficients in rough grids. United States: N. p., 1996. Web. doi:10.1006/jcph.1996.0257.
Shashkov, M, & Steinberg, S. Solving diffusion equations with rough coefficients in rough grids. United States. doi:10.1006/jcph.1996.0257.
Shashkov, M, and Steinberg, S. Sun . "Solving diffusion equations with rough coefficients in rough grids". United States. doi:10.1006/jcph.1996.0257.
@article{osti_530648,
title = {Solving diffusion equations with rough coefficients in rough grids},
author = {Shashkov, M and Steinberg, S},
abstractNote = {A finite-difference algorithm for the numerical solution of diffusion problems in strongly heterogeneous and nonisotropic media is constructed for logically rectangular grids. The performance of this algorithm is comparable to other algorithms for problems with smooth coefficients and regular grids, and it is superior for problems with rough coefficients and/or skewed grids. The algorithm is derived using the support-operators method, which constructs discrete analogs of the divergence and flux operator that satisfy discrete analogs of the important integral identities relating the continuum operators. This paper gives the first application of this method to the solution of diffusion problems in heterogeneous an nonisotropic media. The support-operators method forces the discrete analog of the flux operator to be the negative adjoint of the discrete divergence in an inner product weighted by the conductivity, as in the differential case. Once this is accomplished, many other important properties follow; for example, the scheme is conservative and the discrete analog of the variable material Laplacian is symmetric and negative definite. In addition, on any grid, the discrete divergence is zero on constant vectors and the discrete divergence is zero on constant vectors and the discrete divergence is zero on constant vectors and the discrete flux operator is exact for linear functions in case when K is piecewise constant. Moreover, the discrete gradient`s null space is the constant functions, just as in the continuum. Because the algorithm is flux based, it has twice as many unknowns as more standard algorithms. However, the matrices that need to be inverted are symmetric and positive definite, so the most powerful linear solvers can be applied. Also, the scheme is second-order accurate so, all things considered, it is efficient.},
doi = {10.1006/jcph.1996.0257},
journal = {Journal of Computational Physics},
number = 2,
volume = 129,
place = {United States},
year = {1996},
month = {12}
}