GeM software package for computation of symmetries and conservation laws of differential equations

doi:10.1016/j.cpc.2006.08.001

Computer Physics Communications

Volume 176, Issue 1, 1 January 2007, Pages 48-61

https://doi.org/10.1016/j.cpc.2006.08.001 Get rights and content

Abstract

We present a recently developed Maple-based “GeM” software package for automated symmetry and conservation law analysis of systems of partial and ordinary differential equations (DE). The package contains a collection of powerful easy-to-use routines for mathematicians and applied researchers. A standard program that employs “GeM” routines for symmetry, adjoint symmetry or conservation law analysis of any given DE system occupies several lines of Maple code, and produces output in the canonical form. Classification of symmetries and conservation laws with respect to constitutive functions and parameters present in the given DE system is implemented. The “GeM” package is being successfully used in ongoing research. Run examples include classical and new results.

Program summary

Title of program: GeM

Catalogue identifier: ADYK_v1_0

Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADYK_v1_0

Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland

Licensing provisions: none

Computers: PC-compatible running Maple on MS Windows or Linux; SUN systems running Maple for Unix on OS Solaris

Operating systems under which the program has been tested: Windows 2000, Windows XP, Linux, Solaris

Programming language used: Maple 9.5

Memory required to execute with typical data: below 100 Megabytes

No. of lines in distributed program, including test data, etc.: 4939

No. of bytes in distributed program, including test data, etc.: 166 906

Distribution format: tar.gz

Nature of physical problem: Any physical model containing linear or nonlinear partial or ordinary differential equations.

Method of solution: Symbolic computation of Lie, higher and approximate symmetries by Lie's algorithm. Symbolic computation of conservation laws and adjoint symmetries by using multipliers and Euler operator properties. High performance is achieved by using an efficient representation of the system under consideration and resulting symmetry/conservation law determining equations: all dependent variables and derivatives are represented as symbols rather than functions or expressions.

Restrictions on the complexity of the problem: The GeM module routines are normally able to handle ODE/PDE systems of high orders (up to order seven and possibly higher), depending on the nature of the problem. Classification of symmetries/conservation laws with respect to one or more arbitrary constitutive functions of one or two arguments is normally accomplished successfully.

Typical running time: 1–20 seconds for problems that do not involve classification; 5–1000 seconds for problems that involve classification, depending on complexity.

Introduction

Symmetry, adjoint symmetry and conservation law analysis are practically important means of analyzing nonlinear systems of ordinary and partial differential equations [1], [2].

A symmetry of a system of DEs is any transformation of its solution manifold into itself. One-parameter and multi-parameter Lie groups of point symmetries are found using the general Lie algorithm, which is equally applicable to algebraic, ordinary and partial differential equations. Contact and higher symmetries are computed in a similar algorithmic manner. Closely related is the potential symmetry method [2] for discovering nonlocal symmetries.

Symmetries of ordinary differential equations (ODE) are used for reduction of order and complete integration, as well as for the construction of invariant solutions. Symmetries of partial differential equations (PDE) yield reductions of order and/or number of variables. Invariant (in particular, self-similar) solutions that arise from reduced systems often have transparent physical meaning. Many appropriate examples are found in [3]. Infinite-dimensional symmetry groups are used for the construction of families of new exact solutions from known ones (e.g. [4], [5], [6]). For a nonlinear DE system, from its admitted symmetry group, one can determine whether or not it can be mapped into a linear system by an invertible transformation, and find the explicit form of that transformation [2].

Application of Lie's symmetry method for nontrivial systems often requires extensive algebraic manipulation and the solution of large overdetermined systems of linear PDEs. For many contemporary DE models, especially those that do possess non-trivial symmetry structure, such analysis presents a significant computational challenge. In Section 2.1, we overview the algorithm for symmetry group analysis and related challenges.

An important counterpart to knowledge of the symmetry structure of a PDE system is information about its conservation laws. Conservation laws describe essential physical properties of the modeled process. They are used for the development of appropriate numerical methods and for analysis, in particular, existence, uniqueness and stability analysis (e.g., [7], [8], [9]).

Several algorithms for finding conservation laws of PDEs and PDE systems exist [10]. Such algorithms often involve finding a set of multipliers, such that a linear combination of equations of the given system is a divergence expression. After finding multipliers, fluxes are algorithmically reconstructed (Section 2.2). Sets of symmetries and conservation laws for a given PDE system are closely related (e.g., [11]).

Finding conservation laws and finding symmetries involves similar computational challenges, that consist in reduction of redundant systems of determining equations. Several methods for reduction of large overdetermined systems of partial differential equations have been developed (for a review, see [12]). Many of these methods have been implemented in symbolic software packages. In particular, the Rif package¹ is a part of Waterloo Maple, and is used in the GeM package described in this paper.

A REDUCE-based set of programs by Wolf [13] includes routines for point, contact and higher symmetry, and conservation law/adjoint symmetry analysis of DE systems.

In this work, we introduce a recently developed package GeM for Maple (Section 3). It contains practically useful routines to perform local (Lie, contact, higher, adjoint and approximate) symmetry analysis and search for conservation laws of ODEs and PDEs without human intervention. The use of package routines does not require special knowledge or continuous input. The package is compatible with modern Maple versions (9.5 and above) and thus available for all popular platforms (Linux, Windows, UNIX). Package routines automatically generate determining equations in a form suitable for effective automatic reduction and solution. For DE systems involving constitutive functions or parameters, symmetry/conservation law classification is automatically performed. The module routines handle DE systems of high order and with many equations and many dependent and/or independent variables. Programs that use GeM routines, even ones involving advanced classification, in most cases run effectively on a desktop computer.

Several run examples are presented in the end of the paper; more are available on the program website [14].

Section snippets

The Lie procedure for finding symmetries

Consider a general system of l differential equations of order p $G (x, u, \underset{1}{u}, \dots, \underset{p}{u}) = 0,$ $G = (G^{1}, \dots, G^{l}), x = (x^{1}, \dots, x^{n}) \in X, u = (u^{1}, \dots, u^{m}) \in U,$ $\underset{k}{u} = (\frac{\partial^{k} u^{j}}{\partial x^{i_{1}} \dots \partial x^{i_{k}}} | j = 1, \dots, m) \in U_{k}, k = 1, \dots, p,$ with m dependent and n independent variables. The set of all solutions of (2.1) is a manifold Ω in $(m + n)$ -dimensional space $X \times U$ , which corresponds to a manifold $Ω_{p}$ in the prolonged (jet) space $X \times U \times U_{1} \times \dots \times U_{p}$ of dependent and independent variables together with partial derivatives [1].

Three types of local symmetries of DE systems are

Description

A Waterloo Maple-based package GeM has been recently developed by the author. The package routines are capable of finding local (Lie, contact and higher) symmetries, approximate symmetries in the sense of Fushchich [15], adjoint symmetries and conservation laws of any ODE/PDE system without significant limitations on DE order and number of variables, and without human intervention.

The routines of the module allow the analysis of ODE/PDE systems containing arbitrary constitutive functions and/or

Summary and further remarks

The GeM package for Maple presented in this paper contains a set of automated routines for local (Lie, contact and higher-order) and approximate symmetry analysis and classification, conservation law analysis and classification (through the determination of conservation law multipliers), and adjoint symmetry analysis and classification, for systems of partial and ordinary differential equations. The package routines efficiently produce sets of determining equations that are optimized for

Acknowledgements

The author thanks George Bluman, Stephen Anco, Nataliya Ivanova and the referee of this paper for useful discussion and suggestions, and the National Sciences and Engineering Council of Canada and the Killam Foundation for support.

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^☆: This paper and its associated computer program are available via the Computer Physics Communication homepage on ScienceDirect (http://www.sciencedirect.com/science/journal/00104655).

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GeM software package for computation of symmetries and conservation laws of differential equations☆

Abstract

Program summary

Introduction

Section snippets

The Lie procedure for finding symmetries

Description

Summary and further remarks

Acknowledgements

Phys. Lett. A

Math. Comp. Mod.

Applications of Lie Groups to Differential Equations

Symmetries and Differential Equations

Exact anisotropic MHD equilibria

J. Phys. A

Construction of exact plasma equilibrium solutions with different geometries

Phys. Rev. Lett.

Integrals of nonlinear equations of evolution and solitary waves

Comm. Pure Appl. Math.

The stability of solitary waves

Proc. Roy. Soc. London A

Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity

Arch. Rat. Mech. Anal.