Вычислительная математика - ИСЭМ СО РАН

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О НЕПРЕРЫВНОМ РЕШЕНИИ КРАЕВОЙ ЗАДАЧИ ВОЗНИКАЮЩЕЙ В МО- ДЕЛИРОВАНИИ ПРОЦЕССА ФОРМИРОВАНИЯ ВОЛОКНА ИЗ РАСПЛАВА А.И.Дрегля СИ ГАСИС, Иркутск e-mail: adreglea@gmail.com Аннотация. Рассматривается нелинейное диффернциальное уравнение третьего порядка, возникающее в математической модели формирования волокна из расплава. Для соответствующей краевой задачи доказана теорема существования. Ключевые слова: модель формирования волокна из расплава, механика, нелинейные ОДУ. 207
COMPUTATION OF STABILITY REGION OF SOME BLOCK METHODS AND THEIR APPLICATION TO NUMERICAL SOLUTIONS OF ODES Ming-Gong Lee, Rei-Wei Song, Yu-Hsang Hong Department of Applied Mathematics, Chung-Hua University, Taiwan email: mglee@chu.edu.tw Abstract. Classes of multistage and multistep integration methods which obtain of r new values at each step are studied. Their stability regions were sketched by MATLAB, and their regions are almostA(α)-stable. Their applications to numerical solutions of nonstiff and stiff equations were studied. Key words: Multistage and multistep methods, A(α)-stable, stiff equations. 1. Introduction Numerical solutions for ordinary differential equations (ODEs) have great importance in scientific computation, as they were widely used to model in representing the world. The common methods used to solve these ODEs are categorized as one-step (multistage) methods and multistep (one stage) methods, which Runge-Kutta methods represent the former group, and Adams-Bashforth-Molton method represents the later group. Some multistage are also available in the community. Implicit one-step method has been studied by Stoller and Morrison [5], Butcher [2], and Shampine and Watts [6]. We will consider a class of implicit multistep and multistage methods for solving ordinary differential equations; we can call it block method. It can obtain a block of new values simultaneously which makes this implicit method more competitive. Our aim is to analyze the stability of this Block methods though graphing capability of MATLAB. The interval of stability of a numerical method is especially important in the choice of a method suitable for stiff system. Indeed, for stiff systems, an interval of stability as large as possible is required to avoid a very restricted stepsize h during numerical integration. A large interval of stability may be sufficient in the integration of some stiff ODEs; for example, when the Jacobian matrix of right-hand side function has eigenvalues which are located in a large, narrow strip along the negative axis. Such equations often arise when a second order hyperbolic differential equations in semi-discretized with respect to its space variable [7]. In section 2, we will introduce some known block implicit one-step method used by Shampine, and their numerical implementation by predictor and corrector method. In section 3, we will show the derivation of some of the block multistage/multistep method, and their stability regions will be given at section 4. In section 5, we will give numerical results of these block-type methods by solving some nonstiff and stiff ODEs, and finally section 6 is the conclusion. 2. Block Implicit One-Step Method We wish to approximate the solution of a differential equation, y ′ (x) = f(x, y(x)), y(a) = y a (1) In the interval of [a, b], suppose functionf is continuous and satisfies |f(x, y) − f(x, u)| ≤ L |y − z| (2) on [a, b] × (−∞, ∞) guarantees the existence of a unique solutiony(x) ∈ C 1 [a, b]. 208
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Российская академи
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Russian Academy of Sciences (RAS) R
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СОДЕРЖАНИЕ Абдулли
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ОБЛАСТИ ПРИТЯЖЕНИЯ
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W2,0 S = {x ∈ W1,0 V : f 1 (x)
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при W 0 ≠ ∅ и f 1 W 0 ⊂ G
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Список литературы [
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О ДВУХ ПОДХОДАХ К П
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2. Ниже рассматрива
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то Случай N = 2, L 1 > 0, L
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СРАВНЕНИЕ АЛГОРИТМ
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Совершенно другой
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а) б) 1 0.8 0.6 0.4 0.2 0 −0.2
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ОБ ОДНОМ КЛАССЕ ВЫР
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4. Пучок матриц λA(t, x
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Определение 1. Матр
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Достаточные услови
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Список (17) содержит
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[8] B.E.Cain Real, 3 × 3 D-stable
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12 2 2 3 2 4 2 5 2 = −∆ 2 5 −
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О СВОЙСТВАХ КОНЕЧН
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Из определения 2 сл
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где E ρn ( + λ(E ρn − AA
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где c 1 = (E−A − 0 A 0 )c
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( ) ( ) ( ) x2 − x F 1 (x) = 2 1
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дополнение. Тогда о
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6. Заключение Иссле
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ON THE PROPERTIES OF FINITE-DIMENSI
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где dσ - элемент пло
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Таким образом, сист
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т.е. P - это соприкас
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в которых равномер
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абсолютные значени
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МЕТОД НОРМАЛЬНЫХ С
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значения изображен
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матрицей Грама кан
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узлов на каждой. По
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МЕТОДЫ ИНТЕГРИРОВА
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водить теоретическ
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циально-алгебраиче
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К системе (9) примен
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[4] В.В. Дикуcap. Метод
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где I(x(t)) = ∫ T t 0 a t−t
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Заметим, что (6)-(13) -
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1.4 1.2 1 y 0.8 0.6 0.4 0.2 0 1 2 3
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Потребуем Дифферен
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ON DEVELOPING SYSTEMS MODELS I.V. K
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очень затруднитель
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Теорема 1.3. Пусть пу
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Далее по формулам,
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ФУНДАМЕНТАЛЬНАЯ ОП
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Покажем единственн
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Для завершения док
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где U Nν (t) = 1 ∫ 2πi γ (
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ЧИСЛЕННОЕ РЕШЕНИЕ
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Количественные и к
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Рис. 1: Изменение ск
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THE NUMERICAL SOLUTION FOR ONE PROB
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Определение 2. [1] Со
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Далее нетрудно сос
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Далее подействуем
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поле описывается у
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∑ ∫ (u kl − u 0 kl) (k,l)∈D
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5. Численный экспер
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к тому, что первый п
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умноженную на любо
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1) p < 2 √ r; 2) p 2 √ r. В
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V 1 (z) , доставляющие
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[8] Курош А.Г. Курс вы
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порядка точности п
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где k 1 и k 2 вычисляю
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меньше последнего
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жесткая для явных м
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AN ALGORITHM BASED ON THE SECOND OR
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распознавание, мин
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Пусть A i (δ) - интерв
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Page 172 and 173: [4] Самойлов М. Ю. Опт
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Page 182 and 183: О СВОЙСТВАХ ВЫРОЖД
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Page 186 and 187: [3] Булатов М.В. Числ
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Page 192 and 193: 3. Вычисление форма
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Page 200 and 201: s k = 0, при k < 0 или k > n.
Page 202 and 203: Шаг 2. Найти (оценит
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Page 206 and 207: Obviously function F x (η) is posi
Page 210 and 211: Forh ∈ (0, h 0 ], letx k = a + kh
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Page 214 and 215: 6. Conclusions. A block-type family
Page 216: ТРУДЫ секции "Вычис
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Вычислительная математика - ИСЭМ СО РАН

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