Mimetic difference approximations of partial differential equations

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Goal was to construct local high-order 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 high-order approximations on uniform grids, but are difficult to retain in high-order approximations on nonuniform grids. We also desire local approximations and use only function values at nearby points in the computational ... continued below

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8 p.

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Hyman, J.M.; Shashkov, M.; Staley, M.; Kerr, S.; Steinberg, S. & Castillo, J. August 1, 1997.

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Description

Goal was to construct local high-order 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 high-order approximations on uniform grids, but are difficult to retain in high-order 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 fourth-order local finite-difference 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.

Physical Description

8 p.

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OSTI as DE97008578

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  • Other Information: PBD: [1997]

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  • Other: DE97008578
  • Report No.: LA-UR--97-2174
  • Grant Number: W-7405-ENG-36
  • DOI: 10.2172/518902 | External Link
  • Office of Scientific & Technical Information Report Number: 518902
  • Archival Resource Key: ark:/67531/metadc691762

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  • August 1, 1997

Added to The UNT Digital Library

  • Aug. 14, 2015, 8:43 a.m.

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  • May 20, 2016, 4:26 p.m.

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Hyman, J.M.; Shashkov, M.; Staley, M.; Kerr, S.; Steinberg, S. & Castillo, J. Mimetic difference approximations of partial differential equations, report, August 1, 1997; New Mexico. (digital.library.unt.edu/ark:/67531/metadc691762/: accessed August 19, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.