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First order hyperbolic approach for anisotropic diffusion equation. (English) Zbl 1452.65298

Summary: In this paper, we present a high order finite difference solver for anisotropic diffusion problems based on the first-order hyperbolic system method. In particular, we demonstrate that the construction of a uniformly accurate fifth-order scheme that is independent of the degree of anisotropy is made straightforward by the hyperbolic method with an optimal length scale. We demonstrate that the gradients are computed simultaneously to the same order of accuracy as that of the solution variable by using weight compact finite difference schemes. Furthermore, the approach is extended to improve further the simulation of the magnetized electrons test case previously discussed in [R. Kawashima et al., ibid. 284, 59–69 (2015; Zbl 1351.76315)] and [the first author et al., ibid. 374, 1120–1151 (2018; Zbl 1416.76141)]. Numerical results indicate that these schemes are capable of delivering high accuracy and the proposed approach is expected to allow the hyperbolic method to be successfully applied to a wide variety of linear and nonlinear problems with anisotropic diffusion.

MSC:

65N06 Finite difference methods for boundary value problems involving PDEs
76W05 Magnetohydrodynamics and electrohydrodynamics
76M20 Finite difference methods applied to problems in fluid mechanics
76R50 Diffusion
65Z05 Applications to the sciences

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References:

[1] Braginskii, S., Transport processes in a plasma, Rev. Plasma Phys., 1 (1965)
[2] Perona, P.; Malik, J., Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell., 12, 7, 629-639 (1990)
[3] Basser, P. J.; Jones, D. K., Diffusion-tensor MRI: theory, experimental design and data analysis – a technical review, NMR Biomed., Int. J. Devot. Dev. Appl. Magn. Reson. Vivo, 15, 7-8, 456-467 (2002)
[4] Marchand, R.; Dumberry, M., Carre: a quasi-orthogonal mesh generator for 2D edge plasma modelling, Comput. Phys. Commun., 96, 2-3, 232-246 (1996) · Zbl 0923.76208
[5] Degtyarev, L.; Medvedev, S. Y., Methods for numerical simulation of ideal MHD stability of axisymmetric plasmas, Comput. Phys. Commun., 43, 1, 29-56 (1986) · Zbl 0664.76151
[6] Günter, S.; Yu, Q.; Krüger, J.; Lackner, K., Modelling of heat transport in magnetised plasmas using non-aligned coordinates, J. Comput. Phys., 209, 1, 354-370 (2005) · Zbl 1329.76405
[7] Sovinec, C.; Glasser, A.; Gianakon, T.; Barnes, D.; Nebel, R.; Kruger, S.; Schnack, D.; Plimpton, S.; Tarditi, A.; Chu, M., Nonlinear magnetohydrodynamics simulation using high-order finite elements, J. Comput. Phys., 195, 1, 355-386 (2004) · Zbl 1087.76070
[8] Degond, P.; Deluzet, F.; Negulescu, C., An asymptotic preserving scheme for strongly anisotropic elliptic problems, Multiscale Model. Simul., 8, 2, 645-666 (2010) · Zbl 1190.35216
[9] Degond, P.; Lozinski, A.; Narski, J.; Negulescu, C., An asymptotic-preserving method for highly anisotropic elliptic equations based on a micro – macro decomposition, J. Comput. Phys., 231, 7, 2724-2740 (2012) · Zbl 1332.65165
[10] Mentrelli, A.; Negulescu, C., Asymptotic-preserving scheme for highly anisotropic non-linear diffusion equations, J. Comput. Phys., 231, 24, 8229-8245 (2012)
[11] Chacón, L.; Del-Castillo-Negrete, D.; Hauck, C. D., An asymptotic-preserving semi-Lagrangian algorithm for the time-dependent anisotropic heat transport equation, J. Comput. Phys., 272, 719-746 (2014) · Zbl 1349.82072
[12] van Es, B.; Koren, B.; de Blank, H. J., Finite-difference schemes for anisotropic diffusion, J. Comput. Phys., 272, 526-549 (2014) · Zbl 1349.82136
[13] Kuzmin, D.; Shashkov, M.; Svyatskiy, D., A constrained finite element method satisfying the discrete maximum principle for anisotropic diffusion problems, J. Comput. Phys., 228, 9, 3448-3463 (2009) · Zbl 1163.65085
[14] Sharma, P.; Hammett, G. W., Preserving monotonicity in anisotropic diffusion, J. Comput. Phys., 227, 1, 123-142 (2007) · Zbl 1280.76027
[15] Nishikawa, H., A first-order system approach for diffusion equation, I: second-order residual-distribution schemes, J. Comput. Phys., 227, 1, 315-352 (2007) · Zbl 1127.65062
[16] Nishikawa, H., First-, second-, and third-order finite-volume schemes for diffusion, J. Comput. Phys., 256, 791-805 (2014) · Zbl 1349.65380
[17] Nishikawa, H., First, second, and third order finite-volume schemes for Navier-Stokes equations, (Proc. of 7th AIAA Theoretical Fluid Mechanics Conference (2014), AIAA Aviation and Aeronautics Forum and Exposition 2014: AIAA Aviation and Aeronautics Forum and Exposition 2014 Atlanta, GA), AIAA Paper 2014-2091
[18] Nakashima, Y.; Watanabe, N.; Nishikawa, H., Hyperbolic Navier-Stokes solver for three-dimensional flows, (54th AIAA Aerospace Sciences Meeting. 54th AIAA Aerospace Sciences Meeting, San Diego, CA (2016)), AIAA Paper 2016-1101
[19] Mazaheri, A.; Ricchiuto, M.; Nishikawa, H., A first-order hyperbolic system approach for dispersion, J. Comput. Phys., 321, 593-605 (2016) · Zbl 1349.65323
[20] Baty, H.; Nishikawa, H., Hyperbolic method for magnetic reconnection process in steady state magnetohydrodynamics, Mon. Not. R. Astron. Soc., 459, 624-637 (2016)
[21] Watson, R. A.; Trojak, W.; Tucker, P. G., A simple flux reconstruction approach to solving a Poisson equation to find wall distances for turbulence modelling, (AIAA 2018 Fluid Dynamics Conference. AIAA 2018 Fluid Dynamics Conference, Atlanta, Georgia (2018)), AIAA Paper 2018-4261
[22] Lou, J.; Liu, X.; Luo, H.; Nishikawa, H., Reconstructed discontinuous Galerkin methods for hyperbolic diffusion equations on unstructured grids, (55th AIAA Aerospace Sciences Meeting (2017)), AIAA Paper 310
[23] Lou, J.; Liu, X.; Luo, H.; Nishikawa, H., Reconstructed discontinuous Galerkin methods for hyperbolic diffusion equations on unstructured grids, Commun. Comput. Phys., 25, 5 (2019) · Zbl 1479.76063
[24] Chamarthi, A. S.; Komurasaki, K.; Kawashima, R., High-order upwind and non-oscillatory approach for steady state diffusion, advection – diffusion and application to magnetized electrons, J. Comput. Phys., 374, 1120-1151 (2018) · Zbl 1416.76141
[25] Nishikawa, H.; Nakashima, Y., Dimensional scaling and numerical similarity in hyperbolic method for diffusion, J. Comput. Phys., 355, 121-143 (2018) · Zbl 1380.35151
[26] Lou, J.; Li, L.; Luo, H.; Nishikawa, H., Explicit hyperbolic reconstructed discontinuous Galerkin methods for time-dependent problems, (AIAA 2018 Fluid Dynamics Conference. AIAA 2018 Fluid Dynamics Conference, Atlanta, Georgia (2018)), AIAA Paper 2018-4270
[27] Nishikawa, H.; Nakashima, Y., Dimensional scaling and numerical similarity in hyperbolic method for diffusion, J. Comput. Phys., 355, 121-143 (2018) · Zbl 1380.35151
[28] Liu, Y.; Nishikawa, H., Third-order inviscid and second-order hyperbolic Navier-Stokes solvers for three-dimensional unsteady inviscid and viscous flows, (55th AIAA Aerospace Sciences Meeting. 55th AIAA Aerospace Sciences Meeting, Grapevine, Texas (2017)), AIAA Paper 2017-0738
[29] Nishikawa, H., New-generation hyperbolic Navier-Stokes schemes: \(O(1 / h)\) speed-up and accurate viscous/heat fluxes, (Proc. of 20th AIAA Computational Fluid Dynamics Conference. Proc. of 20th AIAA Computational Fluid Dynamics Conference, Honolulu, Hawaii (2011)), AIAA Paper 2011-3043
[30] Nishikawa, H., On hyperbolic method for diffusion with discontinuous coefficients, J. Comput. Phys., 367, 102-108 (2018) · Zbl 1415.65209
[31] Lou, J.; Li, L.; Luo, H.; Nishikawa, H., First-order hyperbolic system based reconstructed discontinuous Galerkin methods for nonlinear diffusion equations on unstructured grids, (56th AIAA Aerospace Sciences Meeting. 56th AIAA Aerospace Sciences Meeting, Kissimmee, Florida (2018)), AIAA Paper 2018-2094
[32] Li, L.; Lou, J.; Luo, H.; Nishikawa, H., A new formulation of hyperbolic Navier-Stokes solver based on finite volume method on arbitrary grids, (AIAA 2018 Fluid Dynamics Conference. AIAA 2018 Fluid Dynamics Conference, Atlanta, Georgia (2018)), AIAA Paper 2018-4160
[33] Nishikawa, H., First-, second-, and third-order finite-volume schemes for diffusion, J. Comput. Phys., 256, 791-805 (2014) · Zbl 1349.65380
[34] Deng, X.; Zhang, H., Developing high-order weighted compact nonlinear schemes, J. Comput. Phys., 165, 1, 22-44 (2000) · Zbl 0988.76060
[35] Nonomura, T.; Fujii, K., Robust explicit formulation of weighted compact nonlinear scheme, Comput. Fluids, 85, 8-18 (2013) · Zbl 1290.76105
[36] Lele, S. K., Compact finite difference schemes with spectral-like resolution, J. Comput. Phys., 103, 1, 16-42 (1992) · Zbl 0759.65006
[37] Nagarajan, S.; Lele, S. K.; Ferziger, J. H., A robust high-order compact method for large eddy simulation, J. Comput. Phys., 191, 2, 392-419 (2003) · Zbl 1051.76030
[38] Boersma, B. J., A 6th order staggered compact finite difference method for the incompressible Navier-Stokes and scalar transport equations, J. Comput. Phys., 230, 12, 4940-4954 (2011) · Zbl 1416.76172
[39] Jiang, G.-S.; Shu, C.-W., Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126, 126, 202-228 (1995) · Zbl 0877.65065
[40] Deng, X.; Jiang, Y.; Mao, M.; Liu, H.; Li, S.; Tu, G., A family of hybrid cell-edge and cell-node dissipative compact schemes satisfying geometric conservation law, Comput. Fluids, 116, 29-45 (2015) · Zbl 1390.65065
[41] Titarev, V.; Toro, E., Finite-volume WENO schemes for three-dimensional conservation laws, J. Comput. Phys., 201, 1, 238-260 (2004) · Zbl 1059.65078
[42] Nonomura, T.; Morizawa, S.; Terashima, H.; Obayashi, S.; Fujii, K., Numerical (error) issues on compressible multicomponent flows using a high-order differencing scheme: weighted compact nonlinear scheme, J. Comput. Phys., 231, 8, 3181-3210 (2012) · Zbl 1402.76087
[43] Wong, M. L.; Lele, S. K., High-Order Localized Dissipation Weighted Compact Nonlinear Scheme for Shock- and Interface-Capturing in Compressible Flows, vol. 339 (2017), Elsevier Inc. · Zbl 1375.76117
[44] H. Nishikawa, P. Roe, Y. Suzuki, B. van Leer, A general theory of local preconditioning and its application to 2D ideal MHD equations, in: 16th AIAA Computational Fluid Dynamics Conference, p. 3704.; H. Nishikawa, P. Roe, Y. Suzuki, B. van Leer, A general theory of local preconditioning and its application to 2D ideal MHD equations, in: 16th AIAA Computational Fluid Dynamics Conference, p. 3704.
[45] van Es, B.; Koren, B.; de Blank, H. J., Finite-volume scheme for anisotropic diffusion, J. Comput. Phys., 306, 422-442 (2016) · Zbl 1352.65382
[46] Lafleur, T.; Baalrud, S.; Chabert, P., Theory for the anomalous electron transport in hall effect thrusters. ii. Kinetic model, Phys. Plasmas, 23, 5, Article 053503 pp. (2016)
[47] Kawashima, R.; Komurasaki, K.; Schönherr, T., A hyperbolic-equation system approach for magnetized electron fluids in quasi-neutral plasmas, J. Comput. Phys., 284, 59-69 (2015) · Zbl 1351.76315
[48] Tan, S.; Shu, C.-W., Inverse Lax-Wendroff procedure for numerical boundary conditions of conservation laws, J. Comput. Phys., 229, 21, 8144-8166 (2010) · Zbl 1198.65174
[49] Tan, S.; Shu, C. W., A high order moving boundary treatment for compressible inviscid flows, J. Comput. Phys., 230, 15, 6023-6036 (2011) · Zbl 1416.76193
[50] Kawashima, R.; Komurasaki, K.; Schönherr, T., A flux-splitting method for hyperbolic-equation system of magnetized electron fluids in quasi-neutral plasmas, J. Comput. Phys., 310, 202-212 (2016) · Zbl 1349.76485
[51] Nonomura, T.; Iizuka, N.; Fujii, K., Freestream and vortex preservation properties of high-order WENO and WCNS on curvilinear grids, Comput. Fluids, 39, 2, 197-214 (2010) · Zbl 1242.76180
[52] Huynh, H. T., A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods, (18th AIAA Computational Fluid Dynamics Conference (2007)), 4079
[53] Borges, R.; Carmona, M.; Costa, B.; Don, W. S., An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws, J. Comput. Phys., 227, 6, 3191-3211 (2008) · Zbl 1136.65076
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