Spectral stability analysis of the heat conduction equation with convective terms

About the authors:

Rodion M. Skoblikov, Postgraduate Student, University of Tyumen, Tyumen, Russia skoblikov.rodion@yandex.ru, https://orcid.org/0009-0007-2241-246X
Elizaveta P. Kanonirova, Postgraduate Student, University of Tyumen, Tyumen, Russia

stud0000123797@study.utmn.ru, https://orcid.org/0009-0009-7472-2093

Alexander P. Shevelev, Dr. Sci. (Phys.-Math.), Associate Professor, Professor of the Department of Modeling of Physical Processes and Systems, University of Tyumen, Tyumen, Russia; a.p.shevelev@utmn.ru, https://orcid.org/0000-0003-0017-4871

Alexander Ya. Gilmanov, Cand. Sci. (Phys.-Math.), Associate Professor, Department of Modeling of Physical Processes and Systems, School of Natural Sciences, University of Tyumen, Tyumen, Russia; a.y.gilmanov@utmn.ru, https://orcid.org/0000-0002-7115-1629

Alexandra V. Panova, Senior Lecturer, Department of Foreign Languages, Industrial University of Tyumen, Tyumen, Russia; Associate Professor, Department of Modeling of Physical Processes and Systems, University of Tyumen, Tyumen, Russia

panovaav@tyuiu.ru, https://orcid.org/0000-0003-3180-6990

Abstract:

Modeling of physical processes is an important stage in design, predicting, and planning. Numerical methods and the finite difference method in particular, have gained great practical importance, but the latter has limitations. These limitations are related to the requirement of a stable and convergent solution. Stability analysis is a classic method, but the results of such analysis for specific cases have been poorly studied. The aim of the paper is to identify stability conditions for explicit and implicit finite difference schemes of a two-dimensional heat equation with different approximations of convective terms by the Neumann method. Three different types of convective terms’ approximations and their effect on stability conditions are studied: the backward difference, the forward difference, and the central difference. A formalized approach to spectral analysis and finding both sufficient and necessary stability conditions is demonstrated. For each case of approximations, sufficient stability conditions have been obtained that impose strong restrictions on both the dimension of the space-time grid and the scope of specific schemes.

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