The arbitrary order mimetic finite difference method for a diffusion equation with a nonsymmetric diffusion tensor
Here, we present the arbitrary order mimetic finite difference (MFD) discretization for the diffusion equation with nonsymmetric tensorial diffusion coefficient in a mixed formulation on general polygonal meshes. The diffusion tensor is assumed to be positive definite. The asymmetry of the diffusion tensor requires changes to the standard MFD construction. We also present new approach for the construction that guarantees positive definiteness of the nonsymmetric mass matrix in the space of discrete velocities. The numerically observed convergence rate for the scalar quantity matches the predicted one in the case of the lowest order mimetic scheme. For higher orders schemes, we observed superconvergence by one order for the scalar variable which is consistent with the previously published result for a symmetric diffusion tensor. The new scheme was also tested on a timedependent problem modeling the Hall effect in the resistive magnetohydrodynamics.
 Authors:

^{[1]}
;
^{[1]}
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States). Center for Nonlinear Studies (CNLS)
 Publication Date:
 Report Number(s):
 LAUR1627864
Journal ID: ISSN 00219991; TRN: US1800373
 Grant/Contract Number:
 AC5206NA25396
 Type:
 Accepted Manuscript
 Journal Name:
 Journal of Computational Physics
 Additional Journal Information:
 Journal Volume: 348; Journal Issue: C; Journal ID: ISSN 00219991
 Publisher:
 Elsevier
 Research Org:
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Sponsoring Org:
 USDOE Office of Science (SC). Advanced Scientific Computing Research (ASCR) (SC21)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; arbitrary order; general polygonal; general polyhedral; mimetic finite difference; nonsymmetric diffusion; hall effect; resistive magnetohydrodynamics
 OSTI Identifier:
 1412852
Gyrya, V., and Lipnikov, K.. The arbitrary order mimetic finite difference method for a diffusion equation with a nonsymmetric diffusion tensor. United States: N. p.,
Web. doi:10.1016/j.jcp.2017.07.019.
Gyrya, V., & Lipnikov, K.. The arbitrary order mimetic finite difference method for a diffusion equation with a nonsymmetric diffusion tensor. United States. doi:10.1016/j.jcp.2017.07.019.
Gyrya, V., and Lipnikov, K.. 2017.
"The arbitrary order mimetic finite difference method for a diffusion equation with a nonsymmetric diffusion tensor". United States.
doi:10.1016/j.jcp.2017.07.019. https://www.osti.gov/servlets/purl/1412852.
@article{osti_1412852,
title = {The arbitrary order mimetic finite difference method for a diffusion equation with a nonsymmetric diffusion tensor},
author = {Gyrya, V. and Lipnikov, K.},
abstractNote = {Here, we present the arbitrary order mimetic finite difference (MFD) discretization for the diffusion equation with nonsymmetric tensorial diffusion coefficient in a mixed formulation on general polygonal meshes. The diffusion tensor is assumed to be positive definite. The asymmetry of the diffusion tensor requires changes to the standard MFD construction. We also present new approach for the construction that guarantees positive definiteness of the nonsymmetric mass matrix in the space of discrete velocities. The numerically observed convergence rate for the scalar quantity matches the predicted one in the case of the lowest order mimetic scheme. For higher orders schemes, we observed superconvergence by one order for the scalar variable which is consistent with the previously published result for a symmetric diffusion tensor. The new scheme was also tested on a timedependent problem modeling the Hall effect in the resistive magnetohydrodynamics.},
doi = {10.1016/j.jcp.2017.07.019},
journal = {Journal of Computational Physics},
number = C,
volume = 348,
place = {United States},
year = {2017},
month = {7}
}