DOĞRUSAL OLMAYAN ZAMAN RAHATLAMALI BASİTLEŞTİRİLMİŞ MANYETOHİDRODİNAMİKLER İÇİN BACKWARD-EULER YÖNTEMİ ÇÖZÜMLERİ VE KARARLILIK ANALİZİ

Bu çalışmada, Basitleştirilmiş Manyetohidrodinamik (SMHD) denklemlerinin sonlu elemanlar yöntemiyle çözümleri lineer olmayan zaman rahatlatma terimi ile incelenmiştir. SMHD denklemlerine diferansiyel filtre ?(|? − ?̅ |(? − ?̅ )) terimi eklenmiş ve SMHD Lineer Olmayan Zaman Rahatlatma Modeli (SMHDNTRM) tanıtılmıştır. Model, sonlu elemanlar çözümlerinin elde edilmesi için Backward-Euler (BE) yöntemi ile ayrıklaştırılmıştır. Yöntemin kararlılığı da kanıtlanmıştır. Sunulan yöntem koşulsuz olarak kararlıdır. Yöntemin etkinliği, farklı yöntemlerin karşılaştırılmasıyla birkaç örnek ile gösterilmiştir. Tüm hesaplamalar FreeFem++ kullanılarak yapılmıştır.

SOLUTIONS AND STABILITY ANALYSIS OF BACKWARD-EULER METHOD FOR SIMPLIFIED MAGNETOHYDRODYNAMICS WITH NONLINEAR TIME RELAXATION

In this study, the solutions of Simplified Magnetohyrodynamics (SMHD) equations by finite element method are examined with nonlinear time relaxation term. The differential filter κ(|u-u ̅ |(u-u ̅ )) term is added to SMHD equations. Also SMHD Nonlinear Time Relaxation Model (SMHDNTRM) is introduced. The model is discretized by Backward-Euler (BE) method to obtain the finite element solutions. Moreover, the stability of the method is proved. The method is found unconditionally stable. The effectiveness of the method is exemplified by several cases with comparing different methods. FreeFem++ is used for all computations.

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