Mimetic difference approximations of partial differential equations
Abstract
Goal was to construct local highorder difference approximations of differential operators on nonuniform grids that mimic the symmetry properties of the continuum differential operators. Partial differential equations solved with these mimetic difference approximations automatically satisfy discrete versions of conservation laws and analogies to Stoke`s theorem that are true in the continuum and therefore more likely to produce physically faithful results. These symmetries are easily preserved by local discrete highorder approximations on uniform grids, but are difficult to retain in highorder approximations on nonuniform grids. We also desire local approximations and use only function values at nearby points in the computational grid; these methods are especially efficient on computers with distributed memory. We have derived new mimetic fourthorder local finitedifference discretizations of the divergence, gradient, and Laplacian on nonuniform grids. The discrete divergence is the negative of the adjoint of the discrete gradient, and, consequently, the Laplacian is a symmetric negative operator. The new methods derived are local, accurate, reliable, and efficient difference methods that mimic symmetry, conservation, stability, the duality relations and the identities between the gradient, curl, and divergence operators on nonuniform grids. These methods are especially powerful on coarse nonuniform grids and in calculations where the mesh movesmore »
 Authors:

 Los Alamos National Lab., NM (United States)
 New Mexico Univ., Albuquerque, NM (United States)
 San Diego State Univ., CA (United States)
 Publication Date:
 Research Org.:
 Los Alamos National Lab., NM (United States)
 Sponsoring Org.:
 USDOE Assistant Secretary for Human Resources and Administration, Washington, DC (United States)
 OSTI Identifier:
 518902
 Report Number(s):
 LAUR972174
ON: DE97008578
 DOE Contract Number:
 W7405ENG36
 Resource Type:
 Technical Report
 Resource Relation:
 Other Information: PBD: [1997]
 Country of Publication:
 United States
 Language:
 English
 Subject:
 99 MATHEMATICS, COMPUTERS, INFORMATION SCIENCE, MANAGEMENT, LAW, MISCELLANEOUS; PARTIAL DIFFERENTIAL EQUATIONS; NONLINEAR PROBLEMS; MATHEMATICAL OPERATORS; LAPLACIAN; CONSERVATION LAWS; ALGORITHMS
Citation Formats
Hyman, J M, Shashkov, M, Staley, M, Kerr, S, Steinberg, S, and Castillo, J. Mimetic difference approximations of partial differential equations. United States: N. p., 1997.
Web. doi:10.2172/518902.
Hyman, J M, Shashkov, M, Staley, M, Kerr, S, Steinberg, S, & Castillo, J. Mimetic difference approximations of partial differential equations. United States. doi:10.2172/518902.
Hyman, J M, Shashkov, M, Staley, M, Kerr, S, Steinberg, S, and Castillo, J. Fri .
"Mimetic difference approximations of partial differential equations". United States. doi:10.2172/518902. https://www.osti.gov/servlets/purl/518902.
@article{osti_518902,
title = {Mimetic difference approximations of partial differential equations},
author = {Hyman, J M and Shashkov, M and Staley, M and Kerr, S and Steinberg, S and Castillo, J},
abstractNote = {Goal was to construct local highorder difference approximations of differential operators on nonuniform grids that mimic the symmetry properties of the continuum differential operators. Partial differential equations solved with these mimetic difference approximations automatically satisfy discrete versions of conservation laws and analogies to Stoke`s theorem that are true in the continuum and therefore more likely to produce physically faithful results. These symmetries are easily preserved by local discrete highorder approximations on uniform grids, but are difficult to retain in highorder approximations on nonuniform grids. We also desire local approximations and use only function values at nearby points in the computational grid; these methods are especially efficient on computers with distributed memory. We have derived new mimetic fourthorder local finitedifference discretizations of the divergence, gradient, and Laplacian on nonuniform grids. The discrete divergence is the negative of the adjoint of the discrete gradient, and, consequently, the Laplacian is a symmetric negative operator. The new methods derived are local, accurate, reliable, and efficient difference methods that mimic symmetry, conservation, stability, the duality relations and the identities between the gradient, curl, and divergence operators on nonuniform grids. These methods are especially powerful on coarse nonuniform grids and in calculations where the mesh moves to track interfaces or shocks.},
doi = {10.2172/518902},
journal = {},
number = ,
volume = ,
place = {United States},
year = {1997},
month = {8}
}