توصيفگر ها :
روش هاي خطي , روش هاي صريح-ضمني , خاصيت پايداري قوي , خاصيت پايداري خطي , ساخت روش هايي با پايداري قوي
چكيده فارسي :
بسياري از پديده‌ها در علوم رياضي و مهندسي توسط دستگاه‌هاي بزرگ معادلات ديفرانسيل معمولي كه معمولا از گسسته شدن فضا در معادلات ديفرانسيل با مشتقات جزئي بدست مي‌آيند، حل مي‌شوند. چنين دستگاه‌هايي اغلب به‌طور طبيعي از سمت راست به دو قسمت تقسيم مي‌شوند. يكي از اين قسمت‌ها سخت نيست يا ملايم است و قسمت ديگر سخت است. در اين پايان‌نامه حل عددي اين دستگاه‌ها با ادغام روش‌هاي چندمرحله‌اي ضمني قطري صريح-ضمني $(IMEXDIMSIMs)$ به‌عنوان كلاسي از روش‌هاي خطي كلي $(GLM)$ مورد بررسي قرار مي‌گيرد. روش صريح براي قسمتي كه سخت نيست يا ملايم است و روش ضمني براي حل قسمت سخت مناسب است. در اين پايان‌نامه روش‌هايي را خواهيم ساخت كه قسمت صريح داراي ويژگي پايداري قوي و قسمت ضمني $A$-پايدار يا $L$-پايدار باشد. همچنين پايداري اين روش‌ها را هنگامي كه قسمت‌هاي صريح و ضمني با يكديگر تعامل دارند، بررسي مي‌كنيم. به‌طور دقيق‌تر اندازه منطقه پايداري مطلق روش $IMEX$ را زير نظر خواهيم داشت، با فرض اين‌كه بخش ضمني $A$-پايدار يا $L$-پايدار باشد. در نهايت نتيجه آزمايش‌هاي عدد كه روي اين دسته از مسائل اجرا شده‌اند، گزارش شده است.
چكيده انگليسي :
Different methods have been introduced for solving differential equations that usually follow two common approaches. One idea is generalization of the Euler method to multi-step methods and another way is increasing complexity of one-stage methods such as multi-stage Runge-Kutta (RK) methods. General linear methods (GLMs) as an intermediate approach include these methods.
Development and stability analysis of general linear methods have been important area of research in the numerical solution of ordinary differential equations in recent years. Some practical and theoretical aspects of general linear methods are discussed in the book [5] by Butcher, [18] by Hairer and Ernest, and [31] by Hairer and Wanner.
On the other hand, many phenomena in science and engineering are modeled as differential equations that include stiff and non-stiff parts.
Solving these equations, due to their differences in nature, leading to the emergence of methods called explicit-implicit methods (IMEX). In this thesis, using the theoretical framework of general linear methods, IMEXDIMSIMs methods have been investigated, which are suitable methods for solving differential equations with stiff and non-stiff (mild) parts by implicit and explicit methods, respectively. The desired stability properties are obtained for the numerical solution of problems.
In the first chapter, GLMs are introduced and basic concepts are stated about it. Methods that can be written as GLM are given as examples.
Pre-consistency, stage consistency, zero stability are defined and for different methods, conditions for these properties are stated. In the following, the convergence and linear stability of general linear methods are investigated. Finally, the classification of GLM methods that are assigned to stiff or non-stiff systems in successive or parallel computing environments is discussed.
In Chapter 2 IMEXDIMSIM methods are introduced. A - stability and L - stability of the implicit part and also the strong stability of the explicit part have been studied to analyze stability of explicit - implicit methods. In order to improve strong stability property of the proposed methods, the modified IMEXDIMSIMs with order p and stage order q = p are introduced. In this regard, based on [20], a method is presented for constructing IMEXDIMSIMs, which leads to a significant increase of strong stability coefficient, as well as coefficient matrices for methods IMEXDIMSIM2A to IMEXDIMSIM4A and IMEXDISIM2L to IMEXDIMSIM4L.
In the last chapter, to better understand the DIMSIMs, the expansive formulas for method of order p = 2 are written. Also, the results of numerical experiments of DIMSIMs and IMEXDIMSIMs with order p = 2.3 are reported in a table and diagrams related to MATLAB program.
None of the material in this dissertation was mentioned in a specific reference. For more discussion, we have used simultaneously [20] as the main reference and [1], [18], [21], [31], [32] . Book [24] by Jackiewicz is used as a reference to introduce the required methods and preliminaries.
