Yuri A. Blinkov

Involutive bases of polynomial ideals

In this paper we consider an algorithmic technique more general than that proposed by Zharkov and Blinkov for the involutive analysis of polynomial ideals. It is based on a new concept of involutive monomial division which is defined for a monomial set. Such a division provides for each monomial the self-consistent separation of the whole set of variables into two disjoint subsets. They are called multiplicative and non-multiplicative. Given an admissible ordering, this separation is applied to polynomials in terms of their leading monomials. As special cases of the separation we consider those introduced by Janet, Thomas and Pommaret for the purpose of algebraic analysis of partial differential equations. Given involutive division, we define an involutive reduction and an involutive normal form. Then we introduce, in terms of the latter, the concept of involutivity for polynomial systems. We prove that an involutive system is a special, generally redundant, form of a Grobner basis. An algorithm for construction of involutive bases is proposed. It is shown that involutive divisions satisfying certain conditions, for example, those of Janet and Thomas, provide an algorithmic construction of an involutive basis for any polynomial ideal. Some optimization in computation of involutive bases is also analyzed. In particular, we incorporate Buchbergers chain criterion to avoid unnecessary reductions. The implementation for Pommaret division has been done in Reduce.

Construction of Janet Bases I. Monomial Bases

Algorithms for computation of Janet bases for monomial ideals and implementation of these algorithms are presented. As data structures for finite monomial sets the binary trees called Janet trees are selected. An algorithm for construction of a Janet basis for the ideal generated by a finite monomial set is described. This algorithm contains as subalgorithms those to search for Janet divisor in a given tree and to insert monomials into the tree in the process of completion to involution. The algorithms presented have been implemented in C in the form of package for completion of monomial sets to Janet involutive ones. An example is given to illustrate practical efficiency of the monomial algorithms and their implementation.

Grobner Bases and Generation of Difference Schemes for Partial Differential Equations

In this paper we present an algorithmic approach to the generation of fully con- servative 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 ob- tained 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 Grobner 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 Grobner bases and their implementation in Maple. As illustration of the described methods and al- gorithms, we construct a number of difference schemes for Burgers and Falkowich-Karman equations and discuss their numerical properties.

Involution and Difference Schemes for the Navier---Stokes Equations

In the present paper we consider the Navier---Stokes equations for the two-dimensional viscous incompressible fluid flows and apply to these equations our earlier designed general algorithmic approach to generation of finite-difference schemes. In doing so, we complete first the Navier---Stokes equations to involution by computing their Janet basis and discretize this basis by its conversion into the integral conservation law form. Then we again complete the obtained difference system to involution with eliminating the partial derivatives and extracting the minimal Grobner basis from the Janet basis. The elements in the obtained difference Grobner basis that do not contain partial derivatives of the dependent variables compose a conservative difference scheme. By exploiting arbitrariness in the numerical integration approximation we derive two finite-difference schemes that are similar to the classical scheme by Harlow and Welch. Each of the two schemes is characterized by a 5×5 stencil on an orthogonal and uniform grid. We also demonstrate how an inconsistent difference scheme with a 3×3 stencil is generated by an inappropriate numerical approximation of the underlying integrals.

Fast Search for the Janet Divisor

An algorithm for fast search for the involutive monomial Janet divisor is suggested. Such search is an important part of the construction of monomial and polynomial Janet bases. For a data structure for a finite set of monomials, the binary tree is taken, which reflects properties of the Janet division.

Specialized computer algebra system GINV

Computer algebra system GINV (Gröbner INVolutive) is presented. It is designed for studying and solving systems of algebraic, differential, and difference equations of polynomial type by their completion to involution. The system is based on algorithms for constructing involutive Janet and Janet-like bases for polynomial ideals and modules, as well as reduced Gröbner bases, developed by the authors. GINV consists of a library of C++ programs, which is a module of the Python language. It is available at http://invo.jinr.ru/ginv/ and is distributed under the terms of the GPL v2.

Method of Separative Monomials for Involutive Divisions

A method of separative monomials is presented for constructing a partition of a set of monomials associated with a certain involutive division. Given a particular set, the method is capable of constructing a search tree of an involutive divisor with various balancing criteria.

On selection of nonmultiplicative prolongations in computation of Janet bases

We consider three modifications of our basic involutive algorithm for computing polynomial Janet bases. These modifications, which are related to degree-compatible monomial orders, yield specific selection strategies for nonmultiplicative prolongations. Using a standard database of benchmarks designed for testing programs computing Gröbner bases, we compare these algorithmic modifications (in terms of their efficiency) with Faugére’s F4 algorithm, which is built in the Magma computer algebra system.

Generation of difference schemes for the burgers equation by constructing Gröbner bases

A system of basic difference relations approximating an original system of equations, which are required for the generation of difference schemes, is given. The use of the Gröbner basis technique made it possible to generate classes of Lax, Lax-Wendroff, and Godunov schemes for the Burgers equation.

Janet-like gröbner bases

We define a new type of Grobner bases called Janet-like, since their properties are similar to those for Janet bases. In particular, Janet-like bases also admit an explicit formula for the Hilbert function of polynomial ideals. Cardinality of a Janet-like basis never exceeds that of a Janet basis, but in many cases it is substantially less. Especially, Janet-like bases are much more compact than their Janet counterparts when reduced Grobner bases have “sparce” leading monomials sets, e.g., for toric ideals. We present an algorithm for constructing Janet-like bases that is a slight modification of our Janet division algorithm. The former algorithm, by the reason of checking not more but often less number of nonmultiplicative prolongations, is more efficient than the latter one.