The mimetic finite difference method for elliptic and parabolic problems with a staggered discretization of diffusion coefficient

Lipnikov, Konstantin; Manzini, Gianmarco; Moulton, J. David; Shashkov, Mikhail

doi:10.1016/j.jcp.2015.10.031

Title: The mimetic finite difference method for elliptic and parabolic problems with a staggered discretization of diffusion coefficient

Abstract

Numerical schemes for nonlinear parabolic equations based on the harmonic averaging of cell-centered diffusion coefficients break down when some of these coefficients go to zero or their ratio grows. To tackle this problem, we propose new mimetic finite difference schemes that use a staggered discretization of the diffusion coefficient. The primary mimetic operator approximates div (k•); the derived (dual) mimetic operator approximates - ∇(•). The new mimetic schemes preserve symmetry and positive-definiteness of the continuum problem which allows us to use algebraic solvers with optimal complexity. We perform detailed numerical analysis of the new schemes for linear elliptic problems and a specially designed linear parabolic problem that has solution dynamics typical for nonlinear problems. We also show that the new schemes are competitive with the state-of-the-art schemes for steady-state problems but provide much more accurate solution dynamics for the transient problem.

Authors:

Lipnikov, Konstantin ^[1]; Manzini, Gianmarco ^[1]; Moulton, J. David ^[1]; Shashkov, Mikhail ^[1]

Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

Publication Date:: Tue Oct 27 00:00:00 EDT 2015

Research Org.:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Org.:: USDOE National Nuclear Security Administration (NNSA); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

OSTI Identifier:: 1457241

Alternate Identifier(s):: OSTI ID: 1397677

Report Number(s):: LA-UR-15-23755
Journal ID: ISSN 0021-9991; TRN: US1901324

Grant/Contract Number:: AC52-06NA25396

Resource Type:: Journal Article: Accepted Manuscript

Journal Name:: Journal of Computational Physics

Additional Journal Information:: Journal Volume: 305; Journal Issue: C; Journal ID: ISSN 0021-9991

Publisher:: Elsevier

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; Compatible discretizations; Mimetic finite differences; Elliptic and parabolic problems; Unstructured polygonal meshes

Citation Formats


                    Lipnikov, Konstantin, Manzini, Gianmarco, Moulton, J. David, and Shashkov, Mikhail. The mimetic finite difference method for elliptic and parabolic problems with a staggered discretization of diffusion coefficient.  United States: N. p., 2015. 
        Web.  doi:10.1016/j.jcp.2015.10.031.

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                    Lipnikov, Konstantin, Manzini, Gianmarco, Moulton, J. David, & Shashkov, Mikhail. The mimetic finite difference method for elliptic and parabolic problems with a staggered discretization of diffusion coefficient.  United States.  https://doi.org/10.1016/j.jcp.2015.10.031

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                    Lipnikov, Konstantin, Manzini, Gianmarco, Moulton, J. David, and Shashkov, Mikhail. 2015.  
        "The mimetic finite difference method for elliptic and parabolic problems with a staggered discretization of diffusion coefficient".  United States.  https://doi.org/10.1016/j.jcp.2015.10.031.  https://www.osti.gov/servlets/purl/1457241.

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@article{osti_1457241,

  title        = {The mimetic finite difference method for elliptic and parabolic problems with a staggered discretization of diffusion coefficient},

  author       = {Lipnikov, Konstantin and Manzini, Gianmarco and Moulton, J. David and Shashkov, Mikhail},

  abstractNote = {Numerical schemes for nonlinear parabolic equations based on the harmonic averaging of cell-centered diffusion coefficients break down when some of these coefficients go to zero or their ratio grows. To tackle this problem, we propose new mimetic finite difference schemes that use a staggered discretization of the diffusion coefficient. The primary mimetic operator approximates div (k•); the derived (dual) mimetic operator approximates - ∇(•). The new mimetic schemes preserve symmetry and positive-definiteness of the continuum problem which allows us to use algebraic solvers with optimal complexity. We perform detailed numerical analysis of the new schemes for linear elliptic problems and a specially designed linear parabolic problem that has solution dynamics typical for nonlinear problems. We also show that the new schemes are competitive with the state-of-the-art schemes for steady-state problems but provide much more accurate solution dynamics for the transient problem.},

  doi          = {10.1016/j.jcp.2015.10.031},

  url          = {https://www.osti.gov/biblio/1457241},
  journal      = {Journal of Computational Physics},

  issn         = {0021-9991},

  number       = C,

  volume       = 305,

  place        = {United States},

  year         = {Tue Oct 27 00:00:00 EDT 2015},

  month        = {Tue Oct 27 00:00:00 EDT 2015}

}

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Works referenced in this record:

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M-Adaptation in the mimetic finite difference method
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Mimetic finite difference method
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Lipnikov, Konstantin; Manzini, Gianmarco; Shashkov, Mikhail
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Works referencing / citing this record:

Vertex-Centered Linearity-Preserving Schemes for Nonlinear Parabolic Problems on Polygonal Grids
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Wu, Jiming
Journal of Scientific Computing, Vol. 71, Issue 2
https://doi.org/10.1007/s10915-016-0309-3

The High-Order Mixed Mimetic Finite Difference Method for Time-Dependent Diffusion Problems
journal, July 2019

Manzini, Gianmarco; Maguolo, Gianluca; Putti, Mario
Journal of Scientific Computing, Vol. 80, Issue 3
https://doi.org/10.1007/s10915-019-01002-4

An Efficient Hybrid Model for Nonlinear Two-Phase Flow in Fractured Low-Permeability Reservoir
journal, July 2019

Wang, Daigang; Sun, Jingjing; Li, Yong
Energies, Vol. 12, Issue 15
https://doi.org/10.3390/en12152850

A paired spectral-finite difference approach for solving boundary layer flow problems
journal, February 2019

Otegbeye, O.; Motsa, S. S.
Afrika Matematika, Vol. 30, Issue 3-4
https://doi.org/10.1007/s13370-019-00658-3

A Brief Review on Polygonal/Polyhedral Finite Element Methods
journal, October 2018

Perumal, Logah
Mathematical Problems in Engineering, Vol. 2018
https://doi.org/10.1155/2018/5792372

A vertex‐centered and positivity‐preserving scheme for anisotropic diffusion equations on general polyhedral meshes
journal, October 2018

Su, Shuai; Dong, Qiannan; Wu, Jiming
Mathematical Methods in the Applied Sciences, Vol. 42, Issue 1
https://doi.org/10.1002/mma.5324

Numerical artifacts in the Generalized Porous Medium Equation: Why harmonic averaging itself is not to blame
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Maddix, Danielle C.; Sampaio, Luiz; Gerritsen, Margot
Journal of Computational Physics, Vol. 361
https://doi.org/10.1016/j.jcp.2018.02.010

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Title: The mimetic finite difference method for elliptic and parabolic problems with a staggered discretization of diffusion coefficient

Abstract

Citation Formats

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