Вычислительная математика - ИСЭМ СО РАН

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The region is the intersection of three parts which are outside of both black, grey, and light-grey regions. We notice that at Figure 2, the stability region is quite similar to the A(α)-stable stable methods. 5. Numerical Results We will apply the block-type method to one nonstiff and one stiff equation. For the nonstiff equation, the system of equation is given as the following y ′ 1 = y 2y 3 y ′ 2 = −y 1y 3 y ′ 3 = −0.51y 1y 2 with y 1 (0) = 0 y 2 (0) = 1 y 3 (0) = 1 (9) We apply three numerical schemes implicitly, the first one is to apply formula Eq. (3) by Newton iteration, the second is applying a Block formula (Equation (7) by Newton iteration, and the last one is solved by a standard method ode45 from MATLAB. A fixed stepsize h = 5×10 −2 were used at scheme 1 and 2, and variable stepsize strategy scheme 3 was implemented, and their numerical solutions are used as comparison. The outputs of three schemes are given in the following figures. Figure 3 contain solutions of y 1 (x), y 2 (x), solved by three schemes. From the output, the fixed stepsize and variable stepsize strategy do not make too much difference. But y 3 (x), the numerical output show there is some variance, since the initial guess at scheme 1 and 2 may be caused by the calculation of not accurate initial guess. But, Newton iteration did converge back to the correct trace. Figure 3: Numerical output of y 1 (x), y 2 (x), y 3 (x) obtained by schemes 1,2,3. For this nonstiff problem, using P ECE scheme can still solve the problem quite well, the solution is given at Figure 4 But since the stability region will getting smaller [6], and this scheme is not suitable for stiff problem. For the numerical solution of stiff problem, we give the system of equations as the following, y 1 ′ = −0.04y 1 + 10 4 y 2 y 3 y 2 ′ = 0.04y 1 − 10 4 y 2 y 3 − 3 × 10 7 y 2 2 y 3 ′ = 3 × 107 y 2 2 with y 1 (0) = 1 y 2 (0) = 0 y 3 (0) = 0 A fixed stepsize h = 10 −3 were used at scheme 1 and 2, and variable stepsize solver ode15s from MATLAB was implemented, and its numerical solution are used as comparison. Figure 5 contain solutions of y 1 (x), y 2 (x), solved by three schemes. From the output, the fixed stepsize and variable stepsize strategy also do not make too much difference. But for y 2 (x), the numerical output show there is some variance, since the initial guess at scheme 1 and 2 may be caused by the calculation of not accurate initial guess. But, Newton iteration did converge back to the correct trace. But, if we look into the output of y 2 (x), there are some minor differences at the beginning of integration, the Block formula 3.1 solve the solutions with direction variously, that is the reason of using fixed stepsize increment, but eventually the solutions converge back to the right trace as the 211 (10)
Figure 4: Numerical output of y(x) obtained by P ECEof Formula 2.3 Figure 5: Numerical output of y 1 (x), y 2 (x), y 3 (x) obtained by schemes 1,2,3. solutions obtained from ode15s. In the following Figure 6 shows this variance. There is also another special phenomenon, the numerical of oder15s and solutions obtained by Formula (3) are quite similar, they match well at the same trace of the solution curve. The reason maybe the stability region of Formula (3) is a A-stable method. But a similar situation also happens in the nonstiff problem, solutions obtained from Formula (7) match well with ode45 from MATLAB. It shows that the block method maybe similar to a Runge-Kutta method. Details still need more evidence. Figure 6: Variation in numerical outputs y 2 (x)obtained by scheme 1, 2,3 212
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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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как алгоритмы рабо
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где оператор Φ(V, λ)
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Page 172 and 173: [4] Самойлов М. Ю. Опт
Page 174 and 175: K(t) = A + ∫ t 0 k(s)ds. Теор
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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 190 and 191: Обычные интервалы
Page 192 and 193: 3. Вычисление форма
Page 194 and 195: Но rad (GH) |G| · rad H для
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Page 202 and 203: Шаг 2. Найти (оценит
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Page 206 and 207: Obviously function F x (η) is posi
Page 208 and 209: О НЕПРЕРЫВНОМ РЕШЕ
Page 210 and 211: Forh ∈ (0, h 0 ], letx k = a + kh
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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