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TitleGr\"obner bases and generation of difference schemes for partial differential equations.
Author(s) Yuri A. Blinkov, Vladimir P. Gerdt, V.V. Mozzhilkin
TypeArticle in Journal
Abstractn this paper we present an algorithmic approach to the generation of fully conservative difference schemes for linear partial differential equations. The approach is based on enlargement of the equations in their integral conservation law form by extra integral relations between unknown functions and their derivatives, and on discretization of the obtained system. The structure of the discrete system depends on numerical approximation methods for the integrals occurring in the enlarged system. As a result of the discretization, a system of linear polynomial difference equations is derived for the unknown functions and their partial derivatives. A difference scheme is constructed by elimination of all the partial derivatives. The elimination can be achieved by selecting a proper elimination ranking and by computing a Gröbner basis of the linear difference ideal generated by the polynomials in the discrete system. For these purposes we use the difference form of Janet-like Gröbner bases and their implementation in Maple. As illustration of the described methods and algorithms, we construct a number of difference schemes for Burgers and Falkowich-Karman equations and discuss their numerical properties.
Keywordspartial differential equations; conservative difference schemes; difference algebra; linear difference ideal; Gröbner basis; Janet-like basis; computer algebra; Burgers equation; Falkowich-Karman equation
URL http://www.emis.de/journals/SIGMA/2006/Paper051/index.html
JournalSIGMA, Symmetry Integrability Geom. Methods Appl.
Pagespaper 051, 26
PublisherNational Academy of Sciences of Ukraine - NAS Ukraine (Natsional
Translation No
Refereed No