SymDR: Symbol Computer Algebra Library for Generation of Classical and Approximate Dispersion Relations for Systems of Partial Differential Equations

SymDR: Symbol Computer Algebra Library for Generation of Classical and Approximate Dispersion Relations for Systems of Partial Differential Equations Full article

Journal

Lobachevskii Journal of Mathematics
ISSN: 1995-0802 , E-ISSN: 1818-9962

Output data

Year: 2025, Volume: 46, Number: 1, Pages: 1-12 Pages count : 13 DOI: 10.1134/s1995080224608579

Tags

symbolic calculations, SymPy, dispersion relation, approximate dispersion relation, partial differential equation

Authors

Arendarenko M.S. ¹ , Dzhanbekova A.R. ² , Kotov S.V. ¹ , Malyutin M.S. ^3,4 , Savvateeva T.A. ¹ , Samoylov M.V. ¹ , Utyupina V.Y. ¹ , Stoyanovskaya O.P. ⁵

Affiliations

1	Novosibirsk State University, 630090, Novosibirsk, Russia
2	National Research University Higher School of Economics, 101978, Moscow, Russia
3	Federal State Autonomous Educational Institution of Higher Education ‘‘Ural Federal University named after the first President of Russia B.N. Yeltsin’’, 620062, Ekaterinburg, Russia
4	Mikheev Institute of Metal Physics, Ural Branch of the Russian Academy of Sciences, 620108, Ekaterinburg, Russia
5	Lavrentyev Institute of Hydrodynamics, Siberian Branch of the Russian Academy of Sciences, 630090, Novosibirsk, Russia

Mathematical models of numerous processes in continuum mechanics (CM), plasma physics (PP) and astrophysics (AP) are partial differential equations (PDEs). When developing computer models, these equations are replaced by discrete equations that are solved numerically. In order to investigate mathematical and numerical models of CM, PP and AP, the technique of constructing dispersion relations has been developed. Using dispersion relations allows one to derive particular solutions to systems of PDEs, to investigate the stability of solutions for continuous and discrete models, to estimate the order of approximation and the rate of convergence for discrete models, and to establish of the optimal numerical parameters of a discrete model. Dispersion relations describe wave processes (i.e., processes of perturbation transfer with a velocity different from the velocity of matter) in media. The classical dispersion relation is a nonlinear algebraic equation (relating the wave parameters, namely the wave number k and the wave frequency, ω), which corresponds to a continuous system of PDEs. There is a technique that allows one to derive a dispersion relation (classical or approximate, respectively) for a continuous or discrete CM, PP, and AP model. This paper presents a symbolic computer algebra library, developed by the authors, which automates this technique. The current version supports the use of nonstationary models with a single spatial variable both for continuum and finite-difference notation. The library is written in Python using the SymPy symbolic computing package and is available at https://pypi.org/project/symdr/.

Arendarenko M.S. , Dzhanbekova A.R. , Kotov S.V. , Malyutin M.S. , Savvateeva T.A. , Samoylov M.V. , Utyupina V.Y. , Stoyanovskaya O.P.
SymDR: Symbol Computer Algebra Library for Generation of Classical and Approximate Dispersion Relations for Systems of Partial Differential Equations
Lobachevskii Journal of Mathematics. 2025. V.46. N1. P.1-12. DOI: 10.1134/s1995080224608579 WOS Scopus OpenAlex

Submitted:	Sep 1, 2024
Accepted:	Dec 2, 2024
Published print:	May 30, 2025

Web of science:	WOS:001560902800049
Scopus:	2-s2.0-105007086392
OpenAlex:	W4410884024

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