{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T22:27:04Z","timestamp":1775600824323,"version":"3.50.1"},"reference-count":37,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,2,5]],"date-time":"2023-02-05T00:00:00Z","timestamp":1675555200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100006769","name":"Russian Science Foundation","doi-asserted-by":"publisher","award":["22-17-00114"],"award-info":[{"award-number":["22-17-00114"]}],"id":[{"id":"10.13039\/501100006769","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["www.mdpi.com"],"crossmark-restriction":true},"short-container-title":["Mathematics"],"abstract":"<jats:p>We present an algorithm for numerical solution of the equations of magnetohydrodynamics describing the convective dynamo in a plane horizontal layer rotating about an arbitrary axis under geophysically sound boundary conditions. While in many respects we pursue the general approach that was followed by other authors, our main focus is on the accuracy of simulations, especially in the small scales. We employ the Galerkin method. We use products of linear combinations (each involving two to five terms) of Chebyshev polynomials in the vertical Cartesian space variable and Fourier harmonics in the horizontal variables for space discretisation of the toroidal and poloidal potentials of the flow (satisfying the no-slip conditions on the horizontal boundaries) and magnetic field (for which the boundary conditions mimick the presence of a dielectric over the fluid layer and an electrically conducting bottom boundary), and of the deviation of temperature from the steady-state linear profile. For the chosen coefficients in the linear combinations, the products satisfy the respective boundary conditions and constitute non-orthogonal bases in the weighted Lebesgue space. Determining coefficients in the expansion of a given function in such a basis (for instance, for computing the time derivatives of these coefficients) requires solving linear systems of equations for band matrices. Several algorithms for determining the coefficients, which are exploiting algebraically precise relations, have been developed, and their efficiency and accuracy have been numerically investigated for exponentially decaying solutions (encountered when simulating convective regimes which are spatially resolved sufficiently well). For the boundary conditions satisfied by the toroidal component of the flow, our algorithm outperforms the shuttle method, but the latter proves superior when solving the problem for the conditions characterising the poloidal component. While the conditions for the magnetic field on the horizontal boundaries are quite specific, our algorithms for the no-slip boundary conditions are general-purpose and can be applied for treating other boundary-value problems in which the zero value must be admitted on the boundary.<\/jats:p>","DOI":"10.3390\/math11040808","type":"journal-article","created":{"date-parts":[[2023,2,6]],"date-time":"2023-02-06T05:29:05Z","timestamp":1675661345000},"page":"808","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Algorithmic Aspects of Simulation of Magnetic Field Generation by Thermal Convection in a Plane Layer of Fluid"],"prefix":"10.3390","volume":"11","author":[{"given":"Daniil","family":"Tolmachev","sequence":"first","affiliation":[{"name":"Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Sciences, 84\/32 Profsoyuznaya St., 117997 Moscow, Russia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5179-4344","authenticated-orcid":false,"given":"Roman","family":"Chertovskih","sequence":"additional","affiliation":[{"name":"Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, s\/n, 4200-465 Porto, Portugal"}]},{"given":"Vladislav","family":"Zheligovsky","sequence":"additional","affiliation":[{"name":"Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Sciences, 84\/32 Profsoyuznaya St., 117997 Moscow, Russia"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1080\/0309192031000119740","article-title":"A route to magnetic field reversals: An example of an ABC-forced non-linear dynamo","volume":"97","author":"Podvigina","year":"2003","journal-title":"Geophys. Astrophys. Fluid Dyn."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1188","DOI":"10.1016\/j.physd.2010.03.008","article-title":"Dependence of magnetic field generation by thermal convection on the rotation rate: A case study","volume":"239","author":"Chertovskih","year":"2010","journal-title":"Physica D"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1017\/S0022112099007363","article-title":"Convection-driven dynamos in a rotating plane layer","volume":"404","author":"Jones","year":"2000","journal-title":"J. Fluid Mech."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"056308","DOI":"10.1103\/PhysRevE.66.056308","article-title":"Rotating convection-driven dynamos at low Ekman number","volume":"66","author":"Rotvig","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1016\/j.pepi.2013.05.002","article-title":"Model III: Benard convection in the presence of horizontal magnetic field and rotation","volume":"221","author":"Eltayeb","year":"2013","journal-title":"Phys. Earth Planet. Inter."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Zheligovsky, V. (2021). Space analyticity and bounds for derivatives of solutions to the evolutionary equations of diffusive magnetohydrodynamics. Mathematics, 9.","DOI":"10.3390\/math9151789"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"Q05003","DOI":"10.1029\/2007GC001778","article-title":"An efficient spectral method for the simulation of dynamos in Cartesian geometry and its implementation on massively parallel computers","volume":"9","author":"Stellmach","year":"2008","journal-title":"Geochem. Geophys. Geosyst."},{"key":"ref_8","unstructured":"Fox, L., and Parker, I.B. (1968). Chebyshev Polynomials in Numerical Analysis, Oxford University Press."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Canuto, C., Hussaini, M.Y., Quarteroni, A., and Zang, T.A. (1988). Spectral Methods in Fluid Dynamics, Springer.","DOI":"10.1007\/978-3-642-84108-8"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Peyret, R. (2002). Spectral Methods for Incompressible Viscous Flow, Springer.","DOI":"10.1007\/978-1-4757-6557-1"},{"key":"ref_11","first-page":"165","article-title":"Treatment of incompressibility and boundary conditions in 3-D numerical spectral simulations of plane channel flows","volume":"Volume 2","author":"Hirschel","year":"1980","journal-title":"Proceedings of the Third GAMM\u2014Conference on Numerical Methods in Fluid Mechanics"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"260","DOI":"10.1006\/jcph.1995.1163","article-title":"Incompressibility and no-slip boundaries in the Chebyshev-Tau approximation: Correction to Kleiser and Schumann\u2019s influence-matrix solution","volume":"120","author":"Werne","year":"1995","journal-title":"J. Comput. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1016\/S0021-9991(03)00029-9","article-title":"Linearized pipe flow to Reynolds number 107","volume":"186","author":"Meseguer","year":"2003","journal-title":"J. Comput. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"093501","DOI":"10.1103\/PhysRevFluids.6.093501","article-title":"Marginally stable thermal equilibria of Rayleigh\u2013B\u00e9nard convection","volume":"6","author":"Lecoanet","year":"2021","journal-title":"Phys. Rev. Fluids"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"A22","DOI":"10.1017\/jfm.2020.317","article-title":"Invariant states in inclined layer convection. Part 1. Temporal transitions along dynamical connections between invariant states","volume":"898","author":"Reetz","year":"2020","journal-title":"J. Fluid Mech."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"R4","DOI":"10.1017\/jfm.2020.812","article-title":"Steady Rayleigh\u2013B\u00e9nard convection between stress-free boundaries","volume":"905","author":"Wen","year":"2020","journal-title":"J. Fluid Mech."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"R4","DOI":"10.1017\/jfm.2021.1042","article-title":"Steady Rayleigh\u2013B\u00e9nard convection between no-slip boundaries","volume":"933","author":"Wen","year":"2022","journal-title":"J. Fluid Mech."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"033107","DOI":"10.1103\/PhysRevE.101.033107","article-title":"Dynamo action in sliding plates of anisotropic electrical conductivity","volume":"101","author":"Drif","year":"2020","journal-title":"Phys. Rev. E"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"590","DOI":"10.1017\/jfm.2017.286","article-title":"Confinement of rotating convection by a laterally varying magnetic field","volume":"822","author":"Sreenivasan","year":"2017","journal-title":"J. Fluid Mech."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"859","DOI":"10.1016\/j.physleta.2018.12.014","article-title":"Linear stability analysis of the natural convection in inclined rotating parallel plates","volume":"383","author":"Avila","year":"2019","journal-title":"Phys. Lett. A"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Gottlieb, D., and Orszag, S.A. (1977). Numerical Analysis of Spectral Methods: Theory and Applications, SIAM-CBMS.","DOI":"10.1137\/1.9781611970425"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1016\/0021-9991(89)90108-3","article-title":"Divergence-free velocity fields in nonperiodic geometries","volume":"80","author":"Tuckerman","year":"1989","journal-title":"J. Comput. Phys."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"430","DOI":"10.1006\/jcph.2002.6995","article-title":"Exponential time differencing for stiff systems","volume":"176","author":"Cox","year":"2002","journal-title":"J. Comput. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"3300","DOI":"10.1016\/j.physleta.2017.08.025","article-title":"Magnetic field generation by intermittent convection","volume":"381","author":"Chertovskih","year":"2017","journal-title":"Phys. Lett. A"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"14001","DOI":"10.1209\/0295-5075\/112\/14001","article-title":"Route to hyperchaos in Rayleigh\u2013B\u00e9nard convection","volume":"112","author":"Chertovskih","year":"2015","journal-title":"Europhys. Lett."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Jeyabalan, S.R., Chertovskih, R., Gama, S., and Zheligovsky, V. (2022). Nonlinear large-scale perturbations of steady thermal convective dynamo regimes in a plane layer of electrically conducting fluid rotating about the vertical axis. Mathematics, 10.","DOI":"10.3390\/math10162957"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/03091929508228992","article-title":"Equations governing convection in Earth\u2019s core and the geodynamo","volume":"79","author":"Braginsky","year":"1995","journal-title":"Geophys. Astrophys. Fluid Dynamics"},{"key":"ref_28","unstructured":"Boyd, J.P. (2001). Chebyshev and Fourier Spectral Methods, Dover."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1007\/BFb0090349","article-title":"Decomposition of solenoidal fields into poloidal fields, toroidal fields and the mean flow. Applications to the Boussinesq\u2013equations","volume":"Volume 1530","author":"Heywood","year":"1992","journal-title":"The Navier\u2013Stokes Equations II\u2014Theory and Numerical Methods. Proceedings, Oberwolfach 1991"},{"key":"ref_30","unstructured":"Lanczos, C. (1956). Applied Analysis, Prentice-Hall."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1006\/jcph.1993.1068","article-title":"A divergence-free Chebyshev collocation procedure for incompressible flows with two non-periodic directions","volume":"105","author":"Madabhushi","year":"1993","journal-title":"J. Comput. Phys."},{"key":"ref_32","unstructured":"Rivlin, T.J. (1974). Chebyshev Polynomials: From Approximation Theory to Algebra and Number Theory, Wiley-Interscience."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1137\/0916006","article-title":"Efficient spectral-Galerkin method II. Direct solvers of second and fourth order equations by using Chebyshev polynomials","volume":"16","author":"Shen","year":"1995","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_34","unstructured":"Godunov, S.K., and Ryabenkii, V.S. (1987). Difference Schemes. An Introduction to the Underlying Theory, Elsevier. [1st ed.]."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"208","DOI":"10.1016\/j.cam.2005.11.037","article-title":"Effective condition number for finite difference method","volume":"198","author":"Li","year":"2007","journal-title":"J. Comp. Appl. Math."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Mason, J.C., and Handscomb, D.C. (2003). Chebyshev Polynomials, CRC Press.","DOI":"10.1201\/9781420036114"},{"key":"ref_37","unstructured":"Il\u2019in, V.P., Karpov, V.V., and Maslennikov, A.M. (1990). Numerical Methods for Solving Structural Mechanics Problems. A Handbook, Vyshaishaya Shkola."}],"updated-by":[{"DOI":"10.3390\/math11102395","type":"correction","label":"Correction","source":"publisher","updated":{"date-parts":[[2023,2,5]],"date-time":"2023-02-05T00:00:00Z","timestamp":1675555200000}}],"container-title":["Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2227-7390\/11\/4\/808\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,8,3]],"date-time":"2025-08-03T14:30:28Z","timestamp":1754231428000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2227-7390\/11\/4\/808"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,5]]},"references-count":37,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,2]]}},"alternative-id":["math11040808"],"URL":"https:\/\/doi.org\/10.3390\/math11040808","relation":{"correction":[{"id-type":"doi","id":"10.3390\/math11102395","asserted-by":"object"}]},"ISSN":["2227-7390"],"issn-type":[{"value":"2227-7390","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,2,5]]}}}