Zaman Filtresi ile İleri Euler Yönteminin Analizi

Öz

Bu makale, yeni açık Euler yöntemini basit bir zaman filtresi ile güçlendirmenin etkisini incelemektedir. Değişiklik tamamen modülerdir, yalnızca tek bir ek kod satırı gerektirir ve bu nedenle mevcut hesaplama çerçevelerine kolayca dahil edilebilir. Ortaya çıkan şema, basit bir kararlılık analizi sağlar ve hata için anında bir tahmin sunarken, algoritmadaki her aşamanın rolü kavramsal olarak şeffaf kalır. Zaman filtresinin ileri Euler güncellemesine uygulanması, leaping-frog tipi iki aşamalı, ikinci dereceden doğrusal çok adımlı şemaya cebirsel olarak eşdeğer bir yöntem ortaya çıkarır.

Analysis of Forward Euler Method with Time Filter

Öz

This papers examines the impact of augmenting the new explicit Euler method with a simple time filter. The modification is fully modular, requiring only a single additional line of code, and can therefore be incorporated easily into existing computational frameworks. The resulting scheme admits straightforward stability analysis and provides an immediate estimation for error, while the role of each stage in the algorithm remains conceptually transparent. Applying the time filter to the forward Euler update yields a method that is algebraically equivalent to a two-step, second-order linear multistep scheme of leap-frog type

Anahtar Kelimeler

• Time Filter
• Linear Multistep Method
• Modified Equations

Kaynakça

1. Hairer E, Wanner G. Solving ordinary differential equations. II. vol. 14 of Springer Series in Computational Mathematics. Berlin: Springer-Verlag; 2010. Stiff and differential-algebraic problems, Second revised edition. Available from: http://dx.doi.org/10.1007/978-3-642-05221-7.
2. Quarteroni A, Sacco R, Saleri F. Numerical mathematics. vol. 37 of Texts in Applied Mathematics. 2nd ed. Berlin: Springer-Verlag; 2007. Available from: https://doi.org/10.1007/b98885.
3. Atkinson K. An Introduction to Numerical Analysis, 2nd Edition. Wiley; 1991.
4. Guzel A, Trenchea C. The Williams step increases the stability and accuracy of the hoRA time filter. Applied Numerical Mathematics. 2018;131:158-73. doi:https://doi.org/10.1016/j.apnum.2018.05.003.
5. Guzel A, Layton W. Time filters increase accuracy of the fully implicit method. BIT. 2018;58(2):301-15. doi:https://doi.org/10.1007/s10543-018-0695-z.
6. Gu¨zel A. Halving the Error in Second Order Adams-Bashforth Methods via a Simple Time Filter. Black Sea Journal of Engineering and Science. 2026;9:887–893. Available from: https://izlik.org/JA88DK26EJ. doi:10.34248/bsengineering.1870475.
7. Li Y, Trenchea C. Analysis of time filters used with the leapfrog scheme. In: COUPLED VI : proceedings of the VI International Conference on Computational Methods for Coupled Problems in Science and Engineering,Venice, Italy. CIMNE; 2015. p. 1261{1272. Available from: https://hdl.handle.net/2117/192068.
8. Durran DR. Numerical methods for fluid dynamics. vol. 32 of Texts in Applied Mathematics. 2nd ed. New York: Springer; 2010. With applications to geophysics. doi:10.1007/978-1-4419-6412-0.

9. Williams PD. A Proposed Modification to the Robert–Asselin Time Filter. Mon Wea Rev. 2009;137(8):2538-46. doi:10.1175/2009MWR2724.1.
10. Asselin R. Frequency Filter for Time Integrations. Monthly Weather Review. 1972;100(6):487 490. Available from: https://journals.ametsoc.org/view/ journals/mwre/100/6/1520-0493_1972_100_0487_fffti_2_3_co_2.xml. doi:10.1175/1520-0493(1972)100¡0487:FFFTI¿2.3.CO;2.
11. Griffiths DF, Higham DJ. Numerical methods for ordinary differential equations. Springer Undergraduate Mathematics Series. Springer-Verlag London, Ltd., London; 2010. Initial value problems. Available from: http://dx.doi.org/10.1007/978-0-85729-148-6. doi:10.1007/978-0-85729-148-6.