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Algorithm to solve nonlinear Advection Equation numerically using cubic Bspline

خوارزمية لحل معادلة الحمل غير الخطية بطريقة عددية باستخدام توابع B-spline التكعيبية

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 Publication date 2017
and research's language is العربية
 Created by Shamra Editor




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In this paper, we introduce an algorithm to solve the Advection equation by finite element method. In this method, we have chosen Three pattern of cubic B-Spline to approximate the nonlinear solution to convert the nonlinear PDE into a system of ODE, Then we solved this system equation by SSP-RK54 method, And we made a program implementing this algorithm and we checked the program using some examples, which have exact solutions, then we evaluate our results. As a conclusion we found that this method gives accurate results for advection equation.


Artificial intelligence review:
Research summary
تتناول هذه الورقة البحثية تطوير خوارزمية لحل معادلة الحمل غير الخطية باستخدام طريقة العناصر المنتهية وتوابع B-Spline التكعيبية. يتم تحويل معادلة الحمل إلى جملة معادلات تفاضلية عادية باستخدام ثلاثة أنماط من توابع B-Spline التكعيبية، ثم يتم حل هذه الجملة باستخدام طريقة SSP-RK54. تم تنفيذ الخوارزمية على مجموعة من الأمثلة التي لها حلول تحليلية معروفة، وتم حساب الخطأ لتقييم جودة الطريقة. أظهرت النتائج أن الطريقة تعطي حلولاً تقريبية جيدة لمسألة الحمل.
Critical review
دراسة نقدية: تقدم الورقة البحثية حلاً مبتكراً لمعادلة الحمل غير الخطية باستخدام توابع B-Spline التكعيبية، وهو ما يعتبر إضافة قيمة في مجال الحلول العددية للمعادلات التفاضلية الجزئية. ومع ذلك، يمكن تحسين الورقة من خلال تقديم تحليل أعمق للنتائج ومقارنتها بطرق أخرى معروفة في الأدبيات. كما أن الورقة تفتقر إلى مناقشة تفصيلية حول تأثير اختيار المعلمات المختلفة على دقة الحلول. بالإضافة إلى ذلك، يمكن تحسين الورقة من خلال تضمين المزيد من الأمثلة العملية والتطبيقات الواقعية التي يمكن أن تستفيد من هذه الطريقة.
Questions related to the research
  1. ما هي الطريقة المستخدمة في تحويل معادلة الحمل إلى جملة معادلات تفاضلية عادية؟

    تم استخدام ثلاثة أنماط من توابع B-Spline التكعيبية لتحويل معادلة الحمل إلى جملة معادلات تفاضلية عادية.

  2. ما هي الطريقة العددية المستخدمة لحل جملة المعادلات التفاضلية العادية الناتجة؟

    تم استخدام طريقة SSP-RK54 لحل جملة المعادلات التفاضلية العادية الناتجة.

  3. كيف تم تقييم جودة الطريقة المقترحة في الورقة؟

    تم تقييم جودة الطريقة من خلال حساب الخطأ المرتكب عند تطبيق الخوارزمية على مجموعة من الأمثلة التي لها حلول تحليلية معروفة.

  4. ما هي النتائج التي توصلت إليها الورقة بخصوص فعالية الطريقة المقترحة؟

    أظهرت النتائج أن الطريقة تعطي حلولاً تقريبية جيدة لمسألة الحمل، مما يشير إلى فعالية الطريقة المقترحة.


References used
ARORA G, SINGH BK,-2013-Numerical solution of Burgers’ equation with modified cubic B-spline differential quadrature method. Appl Math Comput ,224,166–77
BAKODAH HO, 2016-A Comparative Study of Two Spatial Discretization Schemes for Advection equation. International Journal of Modern Nonlinear Theory and Application, 5, 59-66
BELLMAN R, KASHEF BG, CASTI J,1972- Differential quadrature: a technique for the rapid solution of nonlinear differential equations. J Comput Phy, 10,40–52
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