16 research outputs found
Minimal Involutive Bases
In this paper we present an algorithm for construction of minimal involutive
polynomial bases which are Groebner bases of the special form. The most general
involutive algorithms are based on the concept of involutive monomial division
which leads to partition of variables into multiplicative and
non-multiplicative. This partition gives thereby the self-consistent
computational procedure for constructing an involutive basis by performing
non-multiplicative prolongations and multiplicative reductions. Every specific
involutive division generates a particular form of involutive computational
procedure. In addition to three involutive divisions used by Thomas, Janet and
Pommaret for analysis of partial differential equations we define two new ones.
These two divisions, as well as Thomas division, do not depend on the order of
variables. We prove noetherity, continuity and constructivity of the new
divisions that provides correctness and termination of involutive algorithms
for any finite set of input polynomials and any admissible monomial ordering.
We show that, given an admissible monomial ordering, a monic minimal involutive
basis is uniquely defined and thereby can be considered as canonical much like
the reduced Groebner basis.Comment: 22 page
Gr\"obner Bases and Generation of Difference Schemes for Partial Differential Equations
In 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 Gr\"obner 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 Gr\"obner
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.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
Abstract Minimal involutive bases
In this paper, we present an algorithm for construction of minimal involutive polynomial bases which are GroÈbner bases of the special form. The most general involutive algorithms are based on the concept of involutive monomial division which leads to partition of variables into multiplicative and non-multiplicative. This partition gives thereby the self-consistent computational procedure for constructing an involutive basis by performing non-multiplicative prolongations and multiplicative reductions. Every specific involutive division generates a particular form of involutive computational procedure. In addition to three involutive divisions used by Thomas, Janet and Pommaret for analysis of partial differential equations we define two new ones. These two divisions, as well as Thomas division, do not depend on the order of variables. We prove noetherity, continuity and constructivity of the new divisions that provides correctness and termination of involutive algorithms for any finite set of input polynomials and any admissible monomial ordering. We show that, given an admissible monomial ordering, a monic minimal involutive basis is uniquely defined and thereby can be considered as canonical much like the reduced GroÈbner basis. # 1998 IMACS/Elsevier Science B.V