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Journals / Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics / 2023 / Volume: 72 Issue: 4

Numerical analysis of a time relaxation finite difference method for the heat equation

Özgül İlhan(MUGLA SITKI KOCMAN UNIVERSITY)0000-0003-2199-1301
Osman Raşit Işık(MUGLA SITKI KOCMAN UNIVERSITY, FACULTY OF EDUCATION)0000-0003-1401-4553
Simge Bozkurt(MUGLA SITKI KOCMAN UNIVERSITY, FACULTY OF EDUCATION)0000-0002-6963-7136
Pages
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Publication Date
2023-12-29
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Finally, we apply the method to some test examples.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In this study, we first consider the time-relaxation model, which consists of adding the term $\kappa \left( u-\overline{u}\right) $ to the heat equation.
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Key Findings & Data:Then, an explicit discretization scheme for the model is introduced to find the finite difference solutions.
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Conclusion & Impact:Finally, we apply the method to some test examples.
🔬Design:Modeling & Simulation
📍Scope:Comprehensive Sample
⚙️Variables:Heat equation · time-relaxation model · differential filter

Abstract

In this study, we first consider the time-relaxation model, which consists of adding the term $\kappa \left( u-\overline{u}\right) $ to the heat equation. Then, an explicit discretization scheme for the model is introduced to find the finite difference solutions. We first obtain the solutions by using the scheme and then investigate the method’s consistency, stability, and convergence properties. We prove that the method is consistent and unconditionally stable for any given value of $r$ and appropriate values of $\kappa$ and $\delta$. As a result, the method obtained by adding the time relaxation term to the first-order finite-difference explicit method behaves like the second-order implicit method. Finally, we apply the method to some test examples.

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Cite This Article

İlhan, Ö., Işık, O. R., Bozkurt, S. (2023). Numerical analysis of a time relaxation finite difference method for the heat equation. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, Volume: 72 Issue: 4. https://doi.org/10.31801/cfsuasmas.1243992

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