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    On the Consistency Analysis of Finite Difference Approximations

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    Type
    Article
    Authors
    Michels, Dominik L.
    Gerdt, V. P.
    Blinkov, Yu A.
    Lyakhov, D. A.
    KAUST Department
    Computer Science Program
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Visual Computing Center (VCC)
    Date
    2019-06-24
    Online Publication Date
    2019-06-24
    Print Publication Date
    2019-08
    Embargo End Date
    2020-01-01
    Permanent link to this record
    http://hdl.handle.net/10754/656283
    
    Metadata
    Show full item record
    Abstract
    Finite difference schemes are widely used in applied mathematics to numerically solve partial differential equations. However, for a given solution scheme, it is usually difficult to evaluate the quality of the underlying finite difference approximation with respect to the inheritance of algebraic properties of the differential problem under consideration. In this paper, we present an appropriate quality criterion of strong consistency for finite difference approximations to systems of nonlinear partial differential equations. This property strengthens the standard requirement of consistency of difference equations with differential ones. We use a verification algorithm for strong consistency, which is based on the computation of difference Gröbner bases. This allows for the evaluation and construction of solution schemes that preserve some fundamental algebraic properties of the system at the discrete level. We demonstrate the suggested approach by simulating a Kármán vortex street for the two-dimensional incompressible viscous flow described by the Navier–Stokes equations.
    Citation
    Michels, D. L., Gerdt, V. P., Blinkov, Y. A., & Lyakhov, D. A. (2019). On the Consistency Analysis of Finite Difference Approximations. Journal of Mathematical Sciences, 240(5), 665–677. doi:10.1007/s10958-019-04383-x
    Sponsors
    This work has been partially supported by the King Abdullah University of Science and Technology (KAUST baseline funding), the Russian Foundation for Basic Research (grant 16-01-00080), and the RUDN University Program (5-100).
    Publisher
    Springer New York LLCbarbara.b.bertram@gsk.com
    Journal
    Journal of Mathematical Sciences (United States)
    DOI
    10.1007/s10958-019-04383-x
    Additional Links
    http://link.springer.com/10.1007/s10958-019-04383-x
    ae974a485f413a2113503eed53cd6c53
    10.1007/s10958-019-04383-x
    Scopus Count
    Collections
    Articles; Computer Science Program; Visual Computing Center (VCC); Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

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