{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T14:51:16Z","timestamp":1775055076558,"version":"3.50.1"},"reference-count":20,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2019,2,20]],"date-time":"2019-02-20T00:00:00Z","timestamp":1550620800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>By using symbolic algebraic computation, we construct a strongly-consistent second-order finite difference scheme for steady three-dimensional Stokes flow and a Cartesian solution grid. The scheme has the second order of accuracy and incorporates the pressure Poisson equation. This equation is the integrability condition for the discrete momentum and continuity equations. Our algebraic approach to the construction of difference schemes suggested by the second and the third authors combines the finite volume method, numerical integration, and difference elimination. We make use of the techniques of the differential and difference Janet\/Gr\u00f6bner bases for performing related computations. To prove the strong consistency of the generated scheme, we use these bases to correlate the differential ideal generated by the polynomials in the Stokes equations with the difference ideal generated by the polynomials in the constructed difference scheme. As this takes place, our difference scheme is conservative and inherits permutation symmetry of the differential Stokes flow. For the obtained scheme, we compute the modified differential system and use it to analyze the scheme\u2019s accuracy.<\/jats:p>","DOI":"10.3390\/sym11020269","type":"journal-article","created":{"date-parts":[[2019,2,20]],"date-time":"2019-02-20T11:45:39Z","timestamp":1550663139000},"page":"269","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Algebraic Construction of a Strongly Consistent, Permutationally Symmetric and Conservative Difference Scheme for 3D Steady Stokes Flow"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6145-8774","authenticated-orcid":false,"given":"Xiaojing","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0825-1811","authenticated-orcid":false,"given":"Vladimir","family":"Gerdt","sequence":"additional","affiliation":[{"name":"Laboratory of Information Technologies, Joint Institute for Nuclear Research, Dubna 141980, Russia"},{"name":"Institute of Applied Mathematics &amp; Communications Technology, Peoples\u2019 Friendship University of Russia (RUDN University), Moscow 117198, Russia"}]},{"given":"Yury","family":"Blinkov","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics and Mechanics, Saratov State University, Saratov 413100, Russia"}]}],"member":"1968","published-online":{"date-parts":[[2019,2,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Gerdt, V.P., Koepff, W., Seiler, W.M., and Vorozhtsov, E.V. (2018). A Strongly Consistent Finite Difference Scheme for Steady Stokes Flow and its Modified Equations. Computer Algebra in Scientific Computing, Proceedings of the 20th International Workshop on Computer Algebra in Scientific Computing, Lille, France, 17\u201321 September 2018, Springer.","DOI":"10.1007\/978-3-319-99639-4"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Milne-Tompson, L.M. (1968). Theoretical Hydrodynamics, Macmillan Education LTD. [5th ed.].","DOI":"10.1007\/978-1-349-00517-8"},{"key":"ref_3","unstructured":"Kohr, M., and Pop, I. (2004). Viscous Incompressible Flow for Low Reynolds Numbers, Advances in Boundary Elements, WIT Press."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1039","DOI":"10.1016\/j.jcp.2013.10.015","article-title":"Physics\u2014Compatible numerical methods","volume":"257","author":"Koren","year":"2014","journal-title":"J. Comput. Phys. B"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Strikwerda, J.C. (2004). Finite Difference Schemes and Partial Differential Equations, SIAM. [2nd ed.].","DOI":"10.1137\/1.9780898717938"},{"key":"ref_6","unstructured":"Watt, S.M. (2010). Consistency of finite difference approximations for linear PDE systems and its algorithmic verification. Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, The Association for Computing Machinery."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Adam, G., Bu\u0161a, J., and Hnati\u010d, M. (2012). Consistency analysis of finite difference approximations to PDE Systems. Mathematical Modelling and Computational Science, Proceedings of International Conference on Mathematical Moddeling and Computational Physics, Star\u00e1 Lesn\u00e1, Slovakia, 4\u20138 July 2011, Springer.","DOI":"10.1007\/978-3-642-28212-6"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Levin, A. (2008). Difference Algebra. Algebra and Applications, Springer.","DOI":"10.1007\/978-1-4020-6947-5"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Seiler, W.M. (2010). Involution: The Formal Theory of Differential Equations and its Applications in Computer Algebra. Algorithms and Computation in Mathematics, Springer.","DOI":"10.1007\/978-3-642-01287-7"},{"key":"ref_10","first-page":"051","article-title":"Gr\u00f6bner Bases and Generation of Difference Schemes for Partial Differential Equations","volume":"2","author":"Gerdt","year":"2006","journal-title":"SIGMA"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Gerdt, V.P., Koepff, W., Mayr, E.W., and Vorozhtsov, E.V. (2013). On Consistency of Finite Difference Approximations to the Navier\u2013Stokes Equations. Computer Algebra in Scientific Computing, Proceedings of the 15th International Workshop on Computer Algebra in Scientific Computing, Berlin, Germany, 9\u201313 September 2013, Springer.","DOI":"10.1007\/978-3-319-02297-0"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"408","DOI":"10.1016\/j.amc.2017.06.037","article-title":"Algebraic construction and numerical behavior of a new s-consistent difference scheme for the 2D Navier\u2013Stokes equations","volume":"314","author":"Amodio","year":"2017","journal-title":"Appl. Math. Comput."},{"key":"ref_13","unstructured":"Cojocaru, S., Pfister, G., and Ufnarovski, V. (2004). Involutive algorithms for computing Gr\u00f6bner Bases. Computational Commutative and Non-Commutative Algebraic Geometry, Proceedings of the NATO Advanced Research Workshop on Computational Commutative and Non-Commutative Algebraic Geometry, Chisinau, Moldova, 6\u201311 June 2004, IOS Press."},{"key":"ref_14","unstructured":"Ganzha, V.G., Mayr, E.W., and Vorozhtsov, E.V. (2003). The MAPLE Package Janet: II. Linear Partial Differential Equations. Computer Algebra in Scientific Computing, Proceedings of the 6th International Workshop on Computer Algebra in Scientific Computing, Passau, Germany, 20\u201326 September 2003, Technische Universit\u00e4t."},{"key":"ref_15","first-page":"203","article-title":"Computation of difference Gr\u00f6bner bases","volume":"20","author":"Gerdt","year":"2012","journal-title":"Comput. Sc. J. Moldova"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Shokin, Y.I. (1983). The Method of Differential Approximation, Springer-Verlag.","DOI":"10.1007\/978-3-642-68983-3"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Ganzha, V.G., and Vorozhtsov, E.V. (1996). Computer-aided Analysis of Difference Schemes for Partial Differential Equations, John Wiley & Sons. Inc.","DOI":"10.1002\/9781118032602"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Moin, P. (2010). Fundamentals of Engineering Numerical Analysis, Cambridge University Press. [2nd ed.].","DOI":"10.1017\/CBO9780511781438"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Adams, W.W., and Loustanau, P. (1994). Introduction to Gr\u00f6bner Bases. Graduate Studies in Mathematics, American Mathematical Society.","DOI":"10.1090\/gsm\/003"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1006\/jcph.2001.6754","article-title":"Stability of Pressure Boundary Conditions for Stokes and Navier\u2013Stokes Equations","volume":"172","author":"Petersson","year":"2001","journal-title":"J. Comput. Phys."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/2\/269\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:33:33Z","timestamp":1760186013000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/2\/269"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,2,20]]},"references-count":20,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2019,2]]}},"alternative-id":["sym11020269"],"URL":"https:\/\/doi.org\/10.3390\/sym11020269","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,2,20]]}}}