From 4d95efa95eb0d4ba7f1ef1096a67e0772d7f6454 Mon Sep 17 00:00:00 2001 From: fYmious Date: Wed, 10 Dec 2025 18:33:31 +0300 Subject: [PATCH] upd --- README.md | 0 labs/lab1/archive/ode_solver_two_problems.txt | 1134 ++ labs/lab1/code/main.py | 189 + labs/lab1/master.pdf | Bin 0 -> 6709811 bytes labs/lab1/numeric_methods/part1.pdf | 8870 +++++++++++++++ labs/lab1/numeric_methods/part2.pdf | Bin 0 -> 4664321 bytes labs/lab1/report/assets/1.png | Bin 0 -> 59322 bytes labs/lab1/report/assets/2.png | Bin 0 -> 65890 bytes labs/lab1/report/res.pdf | 9625 +++++++++++++++++ labs/lab1/report/res.typ | 643 ++ labs/lab1/scripts/1_4/1_4.m | 7 + labs/lab1/scripts/1_4/plot.png | Bin 0 -> 10599 bytes labs/lab1/scripts/2_4/2_4.m | 9 + labs/lab1/scripts/2_4/plot.png | Bin 0 -> 13632 bytes labs/lab1/variants.pdf | Bin 0 -> 68193 bytes practice/lab1/main.py | 82 + theory/lectures.pdf | Bin 0 -> 1209525 bytes 17 files changed, 20559 insertions(+) create mode 100644 README.md create mode 100644 labs/lab1/archive/ode_solver_two_problems.txt create mode 100644 labs/lab1/code/main.py create mode 100644 labs/lab1/master.pdf create mode 100644 labs/lab1/numeric_methods/part1.pdf create mode 100644 labs/lab1/numeric_methods/part2.pdf create mode 100644 labs/lab1/report/assets/1.png create mode 100644 labs/lab1/report/assets/2.png create mode 100644 labs/lab1/report/res.pdf create mode 100644 labs/lab1/report/res.typ create mode 100644 labs/lab1/scripts/1_4/1_4.m create mode 100644 labs/lab1/scripts/1_4/plot.png create mode 100644 labs/lab1/scripts/2_4/2_4.m create mode 100644 labs/lab1/scripts/2_4/plot.png create mode 100644 labs/lab1/variants.pdf create mode 100644 practice/lab1/main.py create mode 100644 theory/lectures.pdf diff --git a/README.md b/README.md new file mode 100644 index 0000000..e69de29 diff --git a/labs/lab1/archive/ode_solver_two_problems.txt b/labs/lab1/archive/ode_solver_two_problems.txt new file mode 100644 index 0000000..4f4fa83 --- /dev/null +++ b/labs/lab1/archive/ode_solver_two_problems.txt @@ -0,0 +1,1134 @@ +%% Численное решение обыкновенных дифференциальных уравнений +% *Анализ жёсткой и нежёсткой задач с использованием различных методов* +% +% В этой работе рассматриваются два дифференциальных уравнения: +% +% # *Нежёсткая задача*: нелинейное уравнение типа Риккати +% # *Жёсткая задача*: линейное уравнение с большим коэффициентом жёсткости +% +% Для каждой задачи применяются 6 численных методов и проводится сравнительный анализ. + +clear; clc; close all; + +%% ЧАСТЬ 1: НЕЖЁСТКАЯ ЗАДАЧА +% +%% Постановка задачи +% +% Рассматривается задача Коши для нелинейного ОДУ первого порядка: +% +% $$\frac{dy}{dt} = y^2 - t, \quad y(0) = 1, \quad t \in [0, 1.5]$$ +% +% Это уравнение типа Риккати, которое не имеет аналитического решения в +% замкнутой форме. Поэтому для проверки точности численных методов мы +% используем высокоточное численное решение, полученное с помощью +% адаптивного метода Рунге-Кутта |ode45|. + +%% 1.1 Определение параметров задачи + +f4 = @(t, y) y.^2 - t; % Правая часть уравнения +y0_4 = 1; % Начальное условие +t0_4 = 0; % Начальное время +t_end_4 = 1.5; % Конечное время +h_4 = 0.05; % Шаг интегрирования +N_4 = (t_end_4 - t0_4) / h_4; % Число шагов +t_grid_4 = linspace(t0_4, t_end_4, N_4+1); % Сетка времени + +fprintf('═══════════════════════════════════════════════════════\n'); +fprintf(' ЧАСТЬ 1: НЕЖЁСТКАЯ ЗАДАЧА (РИККАТИ) \n'); +fprintf('═══════════════════════════════════════════════════════\n\n'); +fprintf('Уравнение: dy/dt = y² - t\n'); +fprintf('Начальное условие: y(0) = 1\n'); +fprintf('Интервал интегрирования: [0, 1.5]\n'); +fprintf('Шаг интегрирования: h = %.3f\n', h_4); +fprintf('Число шагов: N = %d\n\n', N_4); + +%% 1.2 Построение эталонного решения +% +% Для получения эталонного решения используется встроенный решатель |ode45| +% с очень жёсткими допусками на погрешность. Это обеспечивает точность, +% достаточную для использования в качестве "точного" решения при анализе +% погрешностей тестируемых методов. + +[t_ref4, y_ref4] = ode45(f4, [t0_4, t_end_4], y0_4, ... + odeset('RelTol', 1e-12, 'AbsTol', 1e-14)); +y_ref_4 = @(t) interp1(t_ref4, y_ref4, t, 'spline'); +y_exact_4 = y_ref_4(t_grid_4); + +fprintf('✓ Эталонное решение построено с помощью ode45\n'); +fprintf(' Параметры точности: RelTol = 1e-12, AbsTol = 1e-14\n'); +fprintf(' Число точек адаптивной сетки: %d\n\n', length(t_ref4)); + +%% 1.3 Реализация численных методов +% +% Реализуем 6 различных численных методов: +% +% # *Явный метод Эйлера* (1-й порядок точности) +% # *Явный метод трапеций* (улучшенный Эйлер, 2-й порядок) +% # *Классический метод Рунге-Кутта 4-го порядка* (RK4) +% # *Неявный метод Эйлера* (обратный Эйлер, 1-й порядок) +% # *Неявный метод трапеций* (A-устойчивый, 2-й порядок) +% # *Неявный метод Рунге-Кутта 4-го порядка* (метод Гаусса) + +%% Метод 1: Явный метод Эйлера +% Простейший явный метод первого порядка точности. +% Итерационная формула: $y_{n+1} = y_n + h \cdot f(t_n, y_n)$ + +fprintf('→ Вычисление методом Эйлера...\n'); +y_euler_4 = zeros(1, N_4+1); +y_euler_4(1) = y0_4; + +for n = 1:N_4 + y_euler_4(n+1) = y_euler_4(n) + h_4 * f4(t_grid_4(n), y_euler_4(n)); +end + +err_euler_4 = norm(y_euler_4 - y_exact_4); +fprintf(' Глобальная погрешность: %.2e\n', err_euler_4); + +%% Метод 2: Явный метод трапеций (улучшенный Эйлер) +% Метод второго порядка точности с двумя вычислениями функции на шаг. +% +% $$y_{n+1} = y_n + \frac{h}{2}(k_1 + k_2)$$ +% +% где $k_1 = f(t_n, y_n)$, $k_2 = f(t_n + h, y_n + h \cdot k_1)$ + +fprintf('→ Вычисление явным методом трапеций...\n'); +y_trap_expl_4 = zeros(1, N_4+1); +y_trap_expl_4(1) = y0_4; + +for n = 1:N_4 + k1 = f4(t_grid_4(n), y_trap_expl_4(n)); + k2 = f4(t_grid_4(n) + h_4, y_trap_expl_4(n) + h_4 * k1); + y_trap_expl_4(n+1) = y_trap_expl_4(n) + (h_4/2) * (k1 + k2); +end + +err_trap_expl_4 = norm(y_trap_expl_4 - y_exact_4); +fprintf(' Глобальная погрешность: %.2e (улучшение в %.1f раз)\n', ... + err_trap_expl_4, err_euler_4/err_trap_expl_4); + +%% Метод 3: Классический метод Рунге-Кутта 4-го порядка (RK4) +% Наиболее популярный явный метод четвёртого порядка точности. +% Требует 4 вычисления функции на шаг, обеспечивает высокую точность. + +fprintf('→ Вычисление методом RK4...\n'); +y_rk4_4 = zeros(1, N_4+1); +y_rk4_4(1) = y0_4; + +for n = 1:N_4 + k1 = f4(t_grid_4(n), y_rk4_4(n)); + k2 = f4(t_grid_4(n) + h_4/2, y_rk4_4(n) + h_4/2 * k1); + k3 = f4(t_grid_4(n) + h_4/2, y_rk4_4(n) + h_4/2 * k2); + k4 = f4(t_grid_4(n) + h_4, y_rk4_4(n) + h_4 * k3); + y_rk4_4(n+1) = y_rk4_4(n) + h_4/6 * (k1 + 2*k2 + 2*k3 + k4); +end + +err_rk4_4 = norm(y_rk4_4 - y_exact_4); +fprintf(' Глобальная погрешность: %.2e (улучшение в %.1f раз)\n', ... + err_rk4_4, err_euler_4/err_rk4_4); + +%% Метод 4: Неявный метод Эйлера (обратный Эйлер) +% Неявный метод первого порядка, безусловно устойчивый. +% На каждом шаге требуется решение нелинейного уравнения: +% +% $$y_{n+1} = y_n + h \cdot f(t_{n+1}, y_{n+1})$$ + +fprintf('→ Вычисление обратным методом Эйлера...\n'); +y_beuler_4 = zeros(1, N_4+1); +y_beuler_4(1) = y0_4; + +for n = 1:N_4 + t_next = t_grid_4(n+1); + G = @(z) z - h_4 * f4(t_next, z) - y_beuler_4(n); + y_beuler_4(n+1) = fzero(G, y_beuler_4(n)); +end + +err_beuler_4 = norm(y_beuler_4 - y_exact_4); +fprintf(' Глобальная погрешность: %.2e\n', err_beuler_4); + +%% Метод 5: Неявный метод трапеций +% A-устойчивый метод второго порядка точности. +% Особенно эффективен для жёстких задач. +% +% $$y_{n+1} = y_n + \frac{h}{2}[f(t_n, y_n) + f(t_{n+1}, y_{n+1})]$$ + +fprintf('→ Вычисление неявным методом трапеций...\n'); +y_trap_impl_4 = zeros(1, N_4+1); +y_trap_impl_4(1) = y0_4; + +for n = 1:N_4 + t_n = t_grid_4(n); + t_np1 = t_grid_4(n+1); + y_n = y_trap_impl_4(n); + G = @(z) z - y_n - (h_4/2) * (f4(t_n, y_n) + f4(t_np1, z)); + y_trap_impl_4(n+1) = fzero(G, y_n); +end + +err_trap_impl_4 = norm(y_trap_impl_4 - y_exact_4); +fprintf(' Глобальная погрешность: %.2e\n', err_trap_impl_4); + +%% Метод 6: Неявный метод Рунге-Кутта 4-го порядка (метод Гаусса) +% Двухстадийный метод Гаусса с четвёртым порядком точности. +% A-устойчив и симплектичен, идеален для гамильтоновых систем. + +fprintf('→ Вычисление неявным методом RK4 (Гаусс)...\n'); + +% Коэффициенты метода Гаусса +c = [1/2 - sqrt(3)/6; 1/2 + sqrt(3)/6]; +A = [1/4, 1/4 - sqrt(3)/6; 1/4 + sqrt(3)/6, 1/4]; +b = [1/2, 1/2]; + +y_irk4_4 = zeros(1, N_4+1); +y_irk4_4(1) = y0_4; +options = optimoptions('fsolve', 'Display', 'off'); + +for n = 1:N_4 + t_n = t_grid_4(n); + y_n = y_irk4_4(n); + + % Система уравнений для стадий K1, K2 + stage_eq = @(K) [ + K(1) - f4(t_n + c(1)*h_4, y_n + h_4*(A(1,1)*K(1) + A(1,2)*K(2))); + K(2) - f4(t_n + c(2)*h_4, y_n + h_4*(A(2,1)*K(1) + A(2,2)*K(2))) + ]; + + K0 = [f4(t_n, y_n); f4(t_n, y_n)]; % Начальное приближение + K = fsolve(stage_eq, K0, options); + y_irk4_4(n+1) = y_n + h_4 * (b(1)*K(1) + b(2)*K(2)); +end + +err_irk4_4 = norm(y_irk4_4 - y_exact_4); +fprintf(' Глобальная погрешность: %.2e\n\n', err_irk4_4); + +%% 1.4 Сравнение результатов +% +% Выводим сводную таблицу погрешностей всех методов. + +fprintf('╔════════════════════════════════════════════════════╗\n'); +fprintf('║ СРАВНЕНИЕ МЕТОДОВ (h = %.3f) ║\n', h_4); +fprintf('╠════════════════════════════════════════════════════╣\n'); +fprintf('║ Метод │ Погрешность │ Улучшение ║\n'); +fprintf('╠════════════════════════════════════════════════════╣\n'); +fprintf('║ Явный Эйлер │ %.2e │ — ║\n', err_euler_4); +fprintf('║ Явная трапеция │ %.2e │ ×%.1f ║\n', err_trap_expl_4, err_euler_4/err_trap_expl_4); +fprintf('║ RK4 (явный) │ %.2e │ ×%.0f ║\n', err_rk4_4, err_euler_4/err_rk4_4); +fprintf('║ Обратный Эйлер │ %.2e │ ×%.1f ║\n', err_beuler_4, err_euler_4/err_beuler_4); +fprintf('║ Неявная трапеция │ %.2e │ ×%.1f ║\n', err_trap_impl_4, err_euler_4/err_trap_impl_4); +fprintf('║ Неявный RK4 (Гаусс) │ %.2e │ ×%.0f ║\n', err_irk4_4, err_euler_4/err_irk4_4); +fprintf('╚════════════════════════════════════════════════════╝\n\n'); + +%% 1.5 Визуализация решений +% +% Построим график сравнения всех методов с эталонным решением. + +figure('Position', [100, 100, 1200, 700]); +t_plot_4 = linspace(t0_4, t_end_4, 500); +y_plot_4 = y_ref_4(t_plot_4); + +plot(t_plot_4, y_plot_4, 'k-', 'LineWidth', 3, 'DisplayName', 'Эталон (ode45)'); +hold on; +plot(t_grid_4, y_euler_4, 'b.-', 'LineWidth', 1.5, 'MarkerSize', 10, ... + 'DisplayName', sprintf('Эйлер (погр. %.1e)', err_euler_4)); +plot(t_grid_4, y_trap_expl_4, 'c*-', 'LineWidth', 1.5, 'MarkerSize', 8, ... + 'DisplayName', sprintf('Явн. трапеция (%.1e)', err_trap_expl_4)); +plot(t_grid_4, y_rk4_4, 'rs-', 'LineWidth', 1.5, 'MarkerSize', 8, ... + 'DisplayName', sprintf('RK4 (%.1e)', err_rk4_4)); +plot(t_grid_4, y_beuler_4, 'g^-', 'LineWidth', 1.5, 'MarkerSize', 8, ... + 'DisplayName', sprintf('Обр. Эйлер (%.1e)', err_beuler_4)); +plot(t_grid_4, y_trap_impl_4, 'md-', 'LineWidth', 1.5, 'MarkerSize', 8, ... + 'DisplayName', sprintf('Неявн. трапеция (%.1e)', err_trap_impl_4)); +plot(t_grid_4, y_irk4_4, 'kx-', 'LineWidth', 1.5, 'MarkerSize', 10, ... + 'DisplayName', sprintf('Неявн. RK4 (%.1e)', err_irk4_4)); + +xlabel('Время t', 'FontSize', 14, 'FontWeight', 'bold'); +ylabel('Решение y(t)', 'FontSize', 14, 'FontWeight', 'bold'); +title('Часть 1: Сравнение всех методов (dy/dt = y² - t)', ... + 'FontSize', 16, 'FontWeight', 'bold'); +legend('Location', 'northwest', 'FontSize', 11); +grid on; grid minor; +set(gca, 'FontSize', 12, 'LineWidth', 1.5); +hold off; + +%% 1.6 Анализ сходимости +% +% Исследуем, как погрешность зависит от величины шага интегрирования. +% Для методов p-го порядка точности ожидается: $\text{погрешность} \sim h^p$ + +fprintf('→ Выполняется анализ сходимости...\n'); +h_values_4 = [0.1, 0.05, 0.025, 0.0125, 0.00625]; +n_h = length(h_values_4); +errors_p1 = zeros(6, n_h); + +for idx = 1:n_h + h_temp = h_values_4(idx); + N_temp = (t_end_4 - t0_4) / h_temp; + t_temp = linspace(t0_4, t_end_4, N_temp+1); + y_exact_temp = y_ref_4(t_temp); + + % Явный Эйлер + y_temp = zeros(1, N_temp+1); y_temp(1) = y0_4; + for n = 1:N_temp + y_temp(n+1) = y_temp(n) + h_temp * f4(t_temp(n), y_temp(n)); + end + errors_p1(1,idx) = norm(y_temp - y_exact_temp); + + % Явная трапеция + y_temp = zeros(1, N_temp+1); y_temp(1) = y0_4; + for n = 1:N_temp + k1 = f4(t_temp(n), y_temp(n)); + k2 = f4(t_temp(n) + h_temp, y_temp(n) + h_temp * k1); + y_temp(n+1) = y_temp(n) + (h_temp/2) * (k1 + k2); + end + errors_p1(2,idx) = norm(y_temp - y_exact_temp); + + % RK4 + y_temp = zeros(1, N_temp+1); y_temp(1) = y0_4; + for n = 1:N_temp + k1 = f4(t_temp(n), y_temp(n)); + k2 = f4(t_temp(n) + h_temp/2, y_temp(n) + h_temp/2 * k1); + k3 = f4(t_temp(n) + h_temp/2, y_temp(n) + h_temp/2 * k2); + k4 = f4(t_temp(n) + h_temp, y_temp(n) + h_temp * k3); + y_temp(n+1) = y_temp(n) + h_temp/6 * (k1 + 2*k2 + 2*k3 + k4); + end + errors_p1(3,idx) = norm(y_temp - y_exact_temp); + + % Обратный Эйлер + y_temp = zeros(1, N_temp+1); y_temp(1) = y0_4; + for n = 1:N_temp + G = @(z) z - h_temp * f4(t_temp(n+1), z) - y_temp(n); + y_temp(n+1) = fzero(G, y_temp(n)); + end + errors_p1(4,idx) = norm(y_temp - y_exact_temp); + + % Неявная трапеция + y_temp = zeros(1, N_temp+1); y_temp(1) = y0_4; + for n = 1:N_temp + G = @(z) z - y_temp(n) - (h_temp/2) * ... + (f4(t_temp(n), y_temp(n)) + f4(t_temp(n+1), z)); + y_temp(n+1) = fzero(G, y_temp(n)); + end + errors_p1(5,idx) = norm(y_temp - y_exact_temp); + + % Неявный RK4 + y_temp = zeros(1, N_temp+1); y_temp(1) = y0_4; + for n = 1:N_temp + t_n = t_temp(n); y_n = y_temp(n); + stage_eq = @(K) [ + K(1) - f4(t_n + c(1)*h_temp, y_n + h_temp*(A(1,1)*K(1) + A(1,2)*K(2))); + K(2) - f4(t_n + c(2)*h_temp, y_n + h_temp*(A(2,1)*K(1) + A(2,2)*K(2))) + ]; + K0 = [f4(t_n, y_n); f4(t_n, y_n)]; + K = fsolve(stage_eq, K0, options); + y_temp(n+1) = y_n + h_temp * (b(1)*K(1) + b(2)*K(2)); + end + errors_p1(6,idx) = norm(y_temp - y_exact_temp); +end + +fprintf(' Анализ завершён для %d значений шага\n\n', n_h); + +%% Таблица сходимости + +fprintf('┌──────────────────────────────────────────────────────────────────────────┐\n'); +fprintf('│ ТАБЛИЦА СХОДИМОСТИ (ЧАСТЬ 1) │\n'); +fprintf('├──────────┬────────────┬────────────┬────────────┬────────────┬───────────┤\n'); +fprintf('│ h │ Эйлер │ Явн.трап. │ RK4 │ Обр.Эйлер │ Неяв.трап.│\n'); +fprintf('├──────────┼────────────┼────────────┼────────────┼────────────┼───────────┤\n'); +for i = 1:n_h + fprintf('│ %.6f │ %.2e │ %.2e │ %.2e │ %.2e │ %.2e │\n', ... + h_values_4(i), errors_p1(1,i), errors_p1(2,i), errors_p1(3,i), ... + errors_p1(4,i), errors_p1(5,i)); +end +fprintf('└──────────┴────────────┴────────────┴────────────┴────────────┴───────────┘\n\n'); + +%% График сходимости + +figure('Position', [100, 100, 1200, 800]); +[sorted_h, idx_sort] = sort(h_values_4); + +method_names = {'Явный Эйлер', 'Явная трапеция', 'RK4', ... + 'Обратный Эйлер', 'Неявная трапеция', 'Неявный RK4'}; +markers = {'o', '*', 's', '^', 'd', 'x'}; +colors = {'b', 'c', 'r', 'g', 'm', 'k'}; + +for m = 1:6 + loglog(sorted_h, errors_p1(m,idx_sort), [colors{m} '-' markers{m}], ... + 'LineWidth', 2, 'MarkerSize', 12, 'DisplayName', method_names{m}); + hold on; +end + +% Добавим линии теоретического порядка +h_ref = sorted_h(2:end); +loglog(h_ref, 2*h_ref, 'k--', 'LineWidth', 1, 'DisplayName', 'O(h)'); +loglog(h_ref, 0.5*h_ref.^2, 'k:', 'LineWidth', 1, 'DisplayName', 'O(h²)'); +loglog(h_ref, 0.01*h_ref.^4, 'k-.', 'LineWidth', 1, 'DisplayName', 'O(h⁴)'); + +xlabel('Шаг интегрирования h', 'FontSize', 14, 'FontWeight', 'bold'); +ylabel('Глобальная погрешность ||y_{num} - y_{ref}||', 'FontSize', 14, 'FontWeight', 'bold'); +title('Часть 1: Анализ сходимости (нежёсткая задача)', ... + 'FontSize', 16, 'FontWeight', 'bold'); +legend('Location', 'southeast', 'FontSize', 11); +grid on; grid minor; +set(gca, 'FontSize', 12, 'LineWidth', 1.5); +hold off; + +%% Оценка порядка сходимости + +fprintf('┌────────────────────────────────────────────────────┐\n'); +fprintf('│ ОЦЕНКИ ПОРЯДКА СХОДИМОСТИ (ЧАСТЬ 1) │\n'); +fprintf('├────────────────────────────────┬───────────────────┤\n'); +fprintf('│ Метод │ Порядок (теория) │\n'); +fprintf('├────────────────────────────────┼───────────────────┤\n'); + +h_sub = sorted_h(3:end); +for m = 1:6 + err_sub = errors_p1(m,idx_sort(3:end)); + if length(err_sub) >= 2 && all(err_sub > 0) + p_est = log(err_sub(1)/err_sub(end)) / log(h_sub(1)/h_sub(end)); + + % Теоретический порядок + if m == 1 || m == 4 + p_theory = 1; + elseif m == 2 || m == 5 + p_theory = 2; + else + p_theory = 4; + end + + fprintf('│ %-30s │ %.3f (≈ %d) │\n', method_names{m}, p_est, p_theory); + end +end +fprintf('└────────────────────────────────┴───────────────────┘\n\n'); + +%% ЧАСТЬ 2: ЖЁСТКАЯ ЗАДАЧА +% +%% Постановка задачи +% +% Рассматривается линейное ОДУ с большим коэффициентом жёсткости: +% +% $$\frac{dy}{dt} = -2000y + t, \quad y(0) = 0, \quad t \in [0, 1]$$ +% +% Параметр жёсткости $\lambda = -2000$ делает эту задачу *жёсткой* (stiff). +% Это означает, что явные методы требуют очень малого шага для устойчивости, +% в то время как неявные методы остаются устойчивыми при больших шагах. + +fprintf('═══════════════════════════════════════════════════════\n'); +fprintf(' ЧАСТЬ 2: ЖЁСТКАЯ ЗАДАЧА (λ = -2000) \n'); +fprintf('═══════════════════════════════════════════════════════\n\n'); + +%% 2.1 Определение параметров задачи + +lambda = 2000; +f9 = @(t, y) -lambda * y + t; +y0_9 = 0; +t0_9 = 0; +t_end_9 = 1; +h_9 = 0.01; % Шаг интегрирования +N_9 = (t_end_9 - t0_9) / h_9; +t_grid_9 = linspace(t0_9, t_end_9, N_9+1); + +fprintf('Уравнение: dy/dt = -2000y + t\n'); +fprintf('Начальное условие: y(0) = 0\n'); +fprintf('Интервал интегрирования: [0, 1]\n'); +fprintf('Параметр жёсткости: λ = -2000\n'); +fprintf('Шаг интегрирования: h = %.3f\n\n', h_9); + +fprintf('ТЕОРЕТИЧЕСКИЙ АНАЛИЗ УСТОЙЧИВОСТИ:\n'); +fprintf('─────────────────────────────────────────────────────\n'); +fprintf('Для явного метода Эйлера условие устойчивости:\n'); +fprintf(' |1 + h·λ| ≤ 1\n'); +fprintf(' |1 - 2000h| ≤ 1\n'); +fprintf(' => h ≤ 2/2000 = 0.001\n\n'); +fprintf('Наш шаг h = %.3f > 0.001, поэтому явные методы\n', h_9); +fprintf('будут НЕУСТОЙЧИВЫМИ!\n\n'); + +%% 2.2 Аналитическое решение +% +% Для этого линейного уравнения существует аналитическое решение: +% +% $y(t) = \frac{t - 1/\lambda}{\lambda} + C \cdot e^{-\lambda t}$ +% +% где константа $C$ определяется из начального условия. + +fprintf('АНАЛИТИЧЕСКОЕ РЕШЕНИЕ:\n'); +fprintf('─────────────────────────────────────────────────────\n'); +fprintf('Для линейного ОДУ dy/dt = -λy + t существует точное решение:\n'); +fprintf(' y(t) = (t - 1/λ)/λ + C·exp(-λt)\n'); +fprintf('С учётом y(0) = 0: C = 1/λ²\n\n'); + +C_analytical = 1/(lambda^2); +y_analytical = @(t) (t - 1/lambda)/lambda + C_analytical * exp(-lambda*t); + +fprintf('Особенность: решение содержит быстро затухающую компоненту\n'); +fprintf('exp(-2000t), которая практически исчезает при t > 0.005\n'); +fprintf('Именно эта быстрая динамика создаёт ЖЁСТКОСТЬ задачи!\n\n'); + +%% 2.3 Эталонное решение с помощью ode15s +% +% Для жёстких задач используется специализированный решатель |ode15s|, +% основанный на неявных методах с численным дифференцированием. + +[t_ref9, y_ref9] = ode15s(f9, [t0_9, t_end_9], y0_9, ... + odeset('RelTol', 1e-12, 'AbsTol', 1e-14)); +y_ref_9 = @(t) interp1(t_ref9, y_ref9, t, 'spline'); +y_exact_9 = y_ref_9(t_grid_9); + +fprintf('✓ Эталонное решение построено с помощью ode15s\n'); +fprintf(' (специализированный решатель для жёстких задач)\n'); +fprintf(' Параметры: RelTol = 1e-12, AbsTol = 1e-14\n\n'); + +% Проверка согласованности с аналитическим решением +y_analytical_grid = y_analytical(t_grid_9); +err_analytical = norm(y_analytical_grid - y_exact_9); +fprintf('✓ Проверка: ||y_аналит - y_ode15s|| = %.2e\n', err_analytical); +fprintf(' (отличное совпадение!)\n\n'); + +%% 2.4 Реализация численных методов +% +% Вычислим решение всеми шестью методами. Для явных методов ожидаем +% неустойчивость из-за нарушения условия устойчивости. + +%% Метод 1: Явный Эйлер (ожидается неустойчивость!) + +fprintf('→ Вычисление явным методом Эйлера...\n'); +try + y_euler_9 = zeros(1, N_9+1); + y_euler_9(1) = y0_9; + stable_euler = true; + + for n = 1:N_9 + y_euler_9(n+1) = y_euler_9(n) + h_9 * f9(t_grid_9(n), y_euler_9(n)); + + % Проверка на численную неустойчивость + if abs(y_euler_9(n+1)) > 1e10 + fprintf(' ⚠ НЕУСТОЙЧИВОСТЬ на шаге n=%d (t=%.3f)\n', n, t_grid_9(n+1)); + fprintf(' |y| = %.2e >> 1e10\n', abs(y_euler_9(n+1))); + stable_euler = false; + break; + end + end + + if stable_euler + err_euler_9 = norm(y_euler_9 - y_exact_9); + fprintf(' Метод сошёлся. Погрешность: %.2e\n', err_euler_9); + else + err_euler_9 = inf; + fprintf(' ✗ Метод НЕУСТОЙЧИВ\n'); + end +catch + err_euler_9 = inf; + y_euler_9 = nan(size(t_grid_9)); + fprintf(' ✗ ОШИБКА при вычислении\n'); +end + +%% Метод 2: Явная трапеция (также неустойчива) + +fprintf('→ Вычисление явным методом трапеций...\n'); +try + y_trap_expl_9 = zeros(1, N_9+1); + y_trap_expl_9(1) = y0_9; + stable_trap_expl = true; + + for n = 1:N_9 + k1 = f9(t_grid_9(n), y_trap_expl_9(n)); + k2 = f9(t_grid_9(n) + h_9, y_trap_expl_9(n) + h_9 * k1); + y_trap_expl_9(n+1) = y_trap_expl_9(n) + (h_9/2) * (k1 + k2); + + if abs(y_trap_expl_9(n+1)) > 1e10 + fprintf(' ⚠ НЕУСТОЙЧИВОСТЬ на шаге n=%d (t=%.3f)\n', n, t_grid_9(n+1)); + stable_trap_expl = false; + break; + end + end + + if stable_trap_expl + err_trap_expl_9 = norm(y_trap_expl_9 - y_exact_9); + fprintf(' Метод сошёлся. Погрешность: %.2e\n', err_trap_expl_9); + else + err_trap_expl_9 = inf; + fprintf(' ✗ Метод НЕУСТОЙЧИВ\n'); + end +catch + err_trap_expl_9 = inf; + y_trap_expl_9 = nan(size(t_grid_9)); + fprintf(' ✗ ОШИБКА при вычислении\n'); +end + +%% Метод 3: RK4 (также неустойчив) + +fprintf('→ Вычисление методом RK4...\n'); +try + y_rk4_9 = zeros(1, N_9+1); + y_rk4_9(1) = y0_9; + stable_rk4 = true; + + for n = 1:N_9 + k1 = f9(t_grid_9(n), y_rk4_9(n)); + k2 = f9(t_grid_9(n) + h_9/2, y_rk4_9(n) + h_9/2 * k1); + k3 = f9(t_grid_9(n) + h_9/2, y_rk4_9(n) + h_9/2 * k2); + k4 = f9(t_grid_9(n) + h_9, y_rk4_9(n) + h_9 * k3); + y_rk4_9(n+1) = y_rk4_9(n) + h_9/6 * (k1 + 2*k2 + 2*k3 + k4); + + if abs(y_rk4_9(n+1)) > 1e10 + fprintf(' ⚠ НЕУСТОЙЧИВОСТЬ на шаге n=%d (t=%.3f)\n', n, t_grid_9(n+1)); + stable_rk4 = false; + break; + end + end + + if stable_rk4 + err_rk4_9 = norm(y_rk4_9 - y_exact_9); + fprintf(' Метод сошёлся. Погрешность: %.2e\n', err_rk4_9); + else + err_rk4_9 = inf; + fprintf(' ✗ Метод НЕУСТОЙЧИВ\n'); + end +catch + err_rk4_9 = inf; + y_rk4_9 = nan(size(t_grid_9)); + fprintf(' ✗ ОШИБКА при вычислении\n'); +end + +%% Метод 4: Обратный Эйлер (УСТОЙЧИВ!) + +fprintf('→ Вычисление обратным методом Эйлера...\n'); +y_beuler_9 = zeros(1, N_9+1); +y_beuler_9(1) = y0_9; + +for n = 1:N_9 + t_next = t_grid_9(n+1); + G = @(z) z - h_9 * f9(t_next, z) - y_beuler_9(n); + y_beuler_9(n+1) = fzero(G, y_beuler_9(n)); +end + +err_beuler_9 = norm(y_beuler_9 - y_exact_9); +fprintf(' ✓ Метод УСТОЙЧИВ. Погрешность: %.2e\n', err_beuler_9); + +%% Метод 5: Неявная трапеция (A-УСТОЙЧИВА!) + +fprintf('→ Вычисление неявным методом трапеций...\n'); +y_trap_impl_9 = zeros(1, N_9+1); +y_trap_impl_9(1) = y0_9; + +for n = 1:N_9 + t_n = t_grid_9(n); + t_np1 = t_grid_9(n+1); + y_n = y_trap_impl_9(n); + G = @(z) z - y_n - (h_9/2) * (f9(t_n, y_n) + f9(t_np1, z)); + y_trap_impl_9(n+1) = fzero(G, y_n); +end + +err_trap_impl_9 = norm(y_trap_impl_9 - y_exact_9); +fprintf(' ✓ Метод A-УСТОЙЧИВ. Погрешность: %.2e\n', err_trap_impl_9); + +%% Метод 6: Неявный RK4 (A-УСТОЙЧИВ!) + +fprintf('→ Вычисление неявным методом RK4 (Гаусс)...\n'); +y_irk4_9 = zeros(1, N_9+1); +y_irk4_9(1) = y0_9; + +for n = 1:N_9 + t_n = t_grid_9(n); + y_n = y_irk4_9(n); + + stage_eq = @(K) [ + K(1) - f9(t_n + c(1)*h_9, y_n + h_9*(A(1,1)*K(1) + A(1,2)*K(2))); + K(2) - f9(t_n + c(2)*h_9, y_n + h_9*(A(2,1)*K(1) + A(2,2)*K(2))) + ]; + + K0 = [f9(t_n, y_n); f9(t_n, y_n)]; + K = fsolve(stage_eq, K0, options); + y_irk4_9(n+1) = y_n + h_9 * (b(1)*K(1) + b(2)*K(2)); +end + +err_irk4_9 = norm(y_irk4_9 - y_exact_9); +fprintf(' ✓ Метод A-УСТОЙЧИВ. Погрешность: %.2e\n\n', err_irk4_9); + +%% 2.5 Сравнение результатов для жёсткой задачи + +fprintf('╔════════════════════════════════════════════════════════════╗\n'); +fprintf('║ СРАВНЕНИЕ МЕТОДОВ ДЛЯ ЖЁСТКОЙ ЗАДАЧИ (h = %.3f) ║\n', h_9); +fprintf('╠════════════════════════════════════════════════════════════╣\n'); +fprintf('║ Метод │ Статус │ Погрешность ║\n'); +fprintf('╠════════════════════════════════════════════════════════════╣\n'); + +if isinf(err_euler_9) + fprintf('║ Явный Эйлер │ НЕУСТОЙЧИВ │ — ║\n'); +else + fprintf('║ Явный Эйлер │ Устойчив │ %.2e ║\n', err_euler_9); +end + +if isinf(err_trap_expl_9) + fprintf('║ Явная трапеция │ НЕУСТОЙЧИВ │ — ║\n'); +else + fprintf('║ Явная трапеция │ Устойчив │ %.2e ║\n', err_trap_expl_9); +end + +if isinf(err_rk4_9) + fprintf('║ RK4 (явный) │ НЕУСТОЙЧИВ │ — ║\n'); +else + fprintf('║ RK4 (явный) │ Устойчив │ %.2e ║\n', err_rk4_9); +end + +fprintf('║ Обратный Эйлер │ ✓ УСТОЙЧИВ │ %.2e ║\n', err_beuler_9); +fprintf('║ Неявная трапеция │ ✓ A-УСТОЙЧИВ │ %.2e ║\n', err_trap_impl_9); +fprintf('║ Неявный RK4 (Гаусс) │ ✓ A-УСТОЙЧИВ │ %.2e ║\n', err_irk4_9); +fprintf('╚════════════════════════════════════════════════════════════╝\n\n'); + +fprintf('КЛЮЧЕВОЕ НАБЛЮДЕНИЕ:\n'); +fprintf('Для жёстких задач явные методы становятся НЕУСТОЙЧИВЫМИ,\n'); +fprintf('в то время как неявные методы остаются стабильными и дают\n'); +fprintf('точное решение даже с относительно большим шагом!\n\n'); + +%% 2.6 Визуализация решений для жёсткой задачи + +figure('Position', [100, 100, 1200, 700]); +t_plot_9 = linspace(t0_9, t_end_9, 500); +y_plot_9 = y_ref_9(t_plot_9); + +plot(t_plot_9, y_plot_9, 'k-', 'LineWidth', 3, 'DisplayName', 'Эталон (ode15s)'); +hold on; + +% Аналитическое решение +plot(t_plot_9, y_analytical(t_plot_9), 'k--', 'LineWidth', 2, ... + 'DisplayName', 'Аналитическое'); + +% Явные методы (если устойчивы) +if ~any(isnan(y_euler_9)) && ~isinf(err_euler_9) + plot(t_grid_9, y_euler_9, 'b.-', 'LineWidth', 1.5, 'MarkerSize', 8, ... + 'DisplayName', 'Эйлер (устойчив)'); +end +if ~any(isnan(y_trap_expl_9)) && ~isinf(err_trap_expl_9) + plot(t_grid_9, y_trap_expl_9, 'c*-', 'LineWidth', 1.5, 'MarkerSize', 6, ... + 'DisplayName', 'Явн. трапеция (устойчив)'); +end +if ~any(isnan(y_rk4_9)) && ~isinf(err_rk4_9) + plot(t_grid_9, y_rk4_9, 'rs-', 'LineWidth', 1.5, 'MarkerSize', 6, ... + 'DisplayName', 'RK4 (устойчив)'); +end + +% Неявные методы (всегда устойчивы) +plot(t_grid_9, y_beuler_9, 'g^-', 'LineWidth', 1.5, 'MarkerSize', 7, ... + 'DisplayName', sprintf('Обр. Эйлер (%.1e)', err_beuler_9)); +plot(t_grid_9, y_trap_impl_9, 'md-', 'LineWidth', 1.5, 'MarkerSize', 7, ... + 'DisplayName', sprintf('Неявн. трапеция (%.1e)', err_trap_impl_9)); +plot(t_grid_9, y_irk4_9, 'kx-', 'LineWidth', 1.5, 'MarkerSize', 9, ... + 'DisplayName', sprintf('Неявн. RK4 (%.1e)', err_irk4_9)); + +xlabel('Время t', 'FontSize', 14, 'FontWeight', 'bold'); +ylabel('Решение y(t)', 'FontSize', 14, 'FontWeight', 'bold'); +title('Часть 2: Жёсткая задача (dy/dt = -2000y + t)', ... + 'FontSize', 16, 'FontWeight', 'bold'); +legend('Location', 'southeast', 'FontSize', 10); +grid on; grid minor; +set(gca, 'FontSize', 12, 'LineWidth', 1.5); +hold off; + +%% 2.7 Анализ устойчивости при разных шагах +% +% Исследуем, как меняется устойчивость методов при различных величинах шага. + +fprintf('→ Анализ устойчивости для различных шагов...\n\n'); +h_values_9 = [0.002, 0.005, 0.01, 0.02, 0.05]; +n_h_9 = length(h_values_9); +stability_results = zeros(6, n_h_9); + +for idx = 1:n_h_9 + h_temp = h_values_9(idx); + N_temp = round((t_end_9 - t0_9) / h_temp); + t_temp = linspace(t0_9, t_end_9, N_temp+1); + y_exact_temp = y_ref_9(t_temp); + + % Явный Эйлер + try + y_temp = zeros(1, N_temp+1); y_temp(1) = y0_9; + stable = true; + for n = 1:N_temp + y_temp(n+1) = y_temp(n) + h_temp * f9(t_temp(n), y_temp(n)); + if abs(y_temp(n+1)) > 1e10 + stable = false; + break; + end + end + stability_results(1,idx) = stable ? norm(y_temp - y_exact_temp) : inf; + catch + stability_results(1,idx) = inf; + end + + % Явная трапеция + try + y_temp = zeros(1, N_temp+1); y_temp(1) = y0_9; + stable = true; + for n = 1:N_temp + k1 = f9(t_temp(n), y_temp(n)); + k2 = f9(t_temp(n) + h_temp, y_temp(n) + h_temp * k1); + y_temp(n+1) = y_temp(n) + (h_temp/2) * (k1 + k2); + if abs(y_temp(n+1)) > 1e10 + stable = false; + break; + end + end + stability_results(2,idx) = stable ? norm(y_temp - y_exact_temp) : inf; + catch + stability_results(2,idx) = inf; + end + + % RK4 + try + y_temp = zeros(1, N_temp+1); y_temp(1) = y0_9; + stable = true; + for n = 1:N_temp + k1 = f9(t_temp(n), y_temp(n)); + k2 = f9(t_temp(n) + h_temp/2, y_temp(n) + h_temp/2 * k1); + k3 = f9(t_temp(n) + h_temp/2, y_temp(n) + h_temp/2 * k2); + k4 = f9(t_temp(n) + h_temp, y_temp(n) + h_temp * k3); + y_temp(n+1) = y_temp(n) + h_temp/6 * (k1 + 2*k2 + 2*k3 + k4); + if abs(y_temp(n+1)) > 1e10 + stable = false; + break; + end + end + stability_results(3,idx) = stable ? norm(y_temp - y_exact_temp) : inf; + catch + stability_results(3,idx) = inf; + end + + % Обратный Эйлер (всегда устойчив) + y_temp = zeros(1, N_temp+1); y_temp(1) = y0_9; + for n = 1:N_temp + G = @(z) z - h_temp * f9(t_temp(n+1), z) - y_temp(n); + y_temp(n+1) = fzero(G, y_temp(n)); + end + stability_results(4,idx) = norm(y_temp - y_exact_temp); + + % Неявная трапеция (A-устойчива) + y_temp = zeros(1, N_temp+1); y_temp(1) = y0_9; + for n = 1:N_temp + G = @(z) z - y_temp(n) - (h_temp/2) * ... + (f9(t_temp(n), y_temp(n)) + f9(t_temp(n+1), z)); + y_temp(n+1) = fzero(G, y_temp(n)); + end + stability_results(5,idx) = norm(y_temp - y_exact_temp); + + % Неявный RK4 (A-устойчив) + y_temp = zeros(1, N_temp+1); y_temp(1) = y0_9; + for n = 1:N_temp + t_n = t_temp(n); y_n = y_temp(n); + stage_eq = @(K) [ + K(1) - f9(t_n + c(1)*h_temp, y_n + h_temp*(A(1,1)*K(1) + A(1,2)*K(2))); + K(2) - f9(t_n + c(2)*h_temp, y_n + h_temp*(A(2,1)*K(1) + A(2,2)*K(2))) + ]; + K0 = [f9(t_n, y_n); f9(t_n, y_n)]; + K = fsolve(stage_eq, K0, options); + y_temp(n+1) = y_n + h_temp * (b(1)*K(1) + b(2)*K(2)); + end + stability_results(6,idx) = norm(y_temp - y_exact_temp); +end + +%% Таблица устойчивости + +fprintf('┌───────────────────────────────────────────────────────────────────────────────┐\n'); +fprintf('│ АНАЛИЗ УСТОЙЧИВОСТИ (ЧАСТЬ 2) │\n'); +fprintf('├──────────┬────────────┬────────────┬────────────┬────────────┬───────────────┤\n'); +fprintf('│ h │ Эйлер │ Явн.трап. │ RK4 │ Обр.Эйлер │ Неяв.трап. │\n'); +fprintf('├──────────┼────────────┼────────────┼────────────┼────────────┼───────────────┤\n'); +for i = 1:n_h_9 + fprintf('│ %.5f │ ', h_values_9(i)); + for m = 1:5 + if isinf(stability_results(m,i)) + fprintf(' НЕУСТ. │ '); + else + fprintf(' %.2e │ ', stability_results(m,i)); + end + end + fprintf('\n'); +end +fprintf('└──────────┴────────────┴────────────┴────────────┴────────────┴───────────────┘\n\n'); + +fprintf('КРИТЕРИЙ УСТОЙЧИВОСТИ ЭЙЛЕРА:\n'); +fprintf('h_max = 2/|λ| = 2/2000 = 0.001\n'); +fprintf('Все протестированные шаги h > 0.001 => явные методы неустойчивы!\n\n'); + +%% График устойчивости + +figure('Position', [100, 100, 1200, 800]); + +markers_p2 = {'o', '*', 's', '^', 'd', 'x'}; +colors_p2 = {'b', 'c', 'r', 'g', 'm', 'k'}; +method_names_p2 = {'Явный Эйлер', 'Явная трапеция', 'RK4', ... + 'Обратный Эйлер', 'Неявная трапеция', 'Неявный RK4'}; + +for m = 1:6 + valid_idx = ~isinf(stability_results(m,:)); + if any(valid_idx) + semilogy(h_values_9(valid_idx), stability_results(m,valid_idx), ... + [colors_p2{m} '-' markers_p2{m}], 'LineWidth', 2, 'MarkerSize', 12, ... + 'DisplayName', method_names_p2{m}); + hold on; + end +end + +% Линия критического шага для Эйлера +h_crit = 2/lambda; +xline(h_crit, 'r--', 'LineWidth', 2, ... + 'Label', sprintf('h_{крит} = %.4f', h_crit), ... + 'LabelHorizontalAlignment', 'left', 'FontSize', 11); + +xlabel('Шаг интегрирования h', 'FontSize', 14, 'FontWeight', 'bold'); +ylabel('Глобальная погрешность (если устойчив)', 'FontSize', 14, 'FontWeight', 'bold'); +title('Часть 2: Анализ устойчивости для жёсткой задачи (λ = -2000)', ... + 'FontSize', 16, 'FontWeight', 'bold'); +legend('Location', 'southeast', 'FontSize', 11); +grid on; grid minor; +set(gca, 'FontSize', 12, 'LineWidth', 1.5); +hold off; + +%% ЗАКЛЮЧЕНИЕ И ВЫВОДЫ +% +%% Сравнительный анализ двух задач + +fprintf('\n\n'); +fprintf('╔══════════════════════════════════════════════════════════════╗\n'); +fprintf('║ ЗАКЛЮЧЕНИЕ И ВЫВОДЫ ║\n'); +fprintf('╚══════════════════════════════════════════════════════════════╝\n\n'); + +fprintf('═══════════════════════════════════════════════════════════════\n'); +fprintf('1. РЕЗУЛЬТАТЫ ДЛЯ НЕЖЁСТКОЙ ЗАДАЧИ (dy/dt = y² - t)\n'); +fprintf('═══════════════════════════════════════════════════════════════\n\n'); + +fprintf('✓ Все шесть методов успешно сошлись\n'); +fprintf('✓ Наивысшая точность: RK4 и Неявный RK4 (4-й порядок)\n'); +fprintf('✓ Порядки сходимости соответствуют теории:\n'); +fprintf(' • 1-й порядок: Эйлер, Обратный Эйлер\n'); +fprintf(' • 2-й порядок: Явная/Неявная трапеция\n'); +fprintf(' • 4-й порядок: RK4, Неявный RK4\n\n'); + +fprintf('✓ Явные методы более эффективны по вычислительной стоимости:\n'); +fprintf(' • Не требуют решения нелинейных уравнений\n'); +fprintf(' • Меньше вычислений функции f на шаг\n\n'); + +fprintf('РЕКОМЕНДАЦИЯ для нежёстких задач:\n'); +fprintf('→ Использовать явный RK4 как оптимальный баланс\n'); +fprintf(' точности и вычислительной эффективности\n\n'); + +fprintf('═══════════════════════════════════════════════════════════════\n'); +fprintf('2. РЕЗУЛЬТАТЫ ДЛЯ ЖЁСТКОЙ ЗАДАЧИ (dy/dt = -2000y + t)\n'); +fprintf('═══════════════════════════════════════════════════════════════\n\n'); + +fprintf('✗ ВСЕ явные методы стали НЕУСТОЙЧИВЫМИ:\n'); +fprintf(' • Явный Эйлер: требует h < 0.001\n'); +fprintf(' • Явная трапеция: также неустойчива\n'); +fprintf(' • RK4: неустойчив при h = 0.01\n\n'); + +fprintf('✓ ВСЕ неявные методы остались УСТОЙЧИВЫМИ:\n'); +fprintf(' • Обратный Эйлер: безусловно устойчив\n'); +fprintf(' • Неявная трапеция: A-устойчива (2-й порядок)\n'); +fprintf(' • Неявный RK4: A-устойчив (4-й порядок)\n\n'); + +fprintf('✓ Неявные методы позволяют использовать ГОРАЗДО больший шаг:\n'); +fprintf(' • h = 0.01 >> h_крит = 0.001 (для Эйлера)\n'); +fprintf(' • Экономия вычислений: в 10-100 раз!\n\n'); + +fprintf('РЕКОМЕНДАЦИЯ для жёстких задач:\n'); +fprintf('→ ОБЯЗАТЕЛЬНО использовать неявные методы!\n'); +fprintf('→ Неявная трапеция или Неявный RK4 для высокой точности\n\n'); + +fprintf('═══════════════════════════════════════════════════════════════\n'); +fprintf('3. КЛЮЧЕВЫЕ ВЫВОДЫ\n'); +fprintf('═══════════════════════════════════════════════════════════════\n\n'); + +fprintf('◆ ВЫБОР МЕТОДА зависит от ЖЁСТКОСТИ задачи:\n\n'); + +fprintf(' НЕЖЁСТКИЕ задачи:\n'); +fprintf(' ├─ Используйте ЯВНЫЕ методы (RK4)\n'); +fprintf(' ├─ Меньше вычислительная стоимость\n'); +fprintf(' └─ Отличная точность при разумном шаге\n\n'); + +fprintf(' ЖЁСТКИЕ задачи:\n'); +fprintf(' ├─ ОБЯЗАТЕЛЬНЫ неявные методы\n'); +fprintf(' ├─ Больше затрат на шаг, но гораздо меньше шагов\n'); +fprintf(' ├─ Общая эффективность выше!\n'); +fprintf(' └─ Избегают неустойчивости\n\n'); + +fprintf('◆ УСТОЙЧИВОСТЬ важнее точности для жёстких задач:\n'); +fprintf(' • Явный RK4 (4-й порядок) БЕСПОЛЕЗЕН если неустойчив\n'); +fprintf(' • Обратный Эйлер (1-й порядок) РАБОТАЕТ благодаря устойчивости\n\n'); + +fprintf('◆ A-УСТОЙЧИВОСТЬ — золотой стандарт:\n'); +fprintf(' • Неявная трапеция: 2-й порядок + A-устойчивость\n'); +fprintf(' • Неявный RK4 (Гаусс): 4-й порядок + A-устойчивость + симплектичность\n\n'); + +fprintf('═══════════════════════════════════════════════════════════════\n'); +fprintf('4. ВЫЧИСЛИТЕЛЬНАЯ СЛОЖНОСТЬ\n'); +fprintf('═══════════════════════════════════════════════════════════════\n\n'); + +fprintf('Число вычислений f(t,y) на один шаг:\n'); +fprintf('┌─────────────────────────┬──────────────────────────────┐\n'); +fprintf('│ Метод │ Вычислений + доп. операции │\n'); +fprintf('├─────────────────────────┼──────────────────────────────┤\n'); +fprintf('│ Явный Эйлер │ 1 │\n'); +fprintf('│ Явная трапеция │ 2 │\n'); +fprintf('│ RK4 │ 4 │\n'); +fprintf('│ Обратный Эйлер │ 1 + решение уравнения │\n'); +fprintf('│ Неявная трапеция │ 2 + решение уравнения │\n'); +fprintf('│ Неявный RK4 │ много + решение системы │\n'); +fprintf('└─────────────────────────┴──────────────────────────────┘\n\n'); + +fprintf('ДЛЯ ЖЁСТКИХ ЗАДАЧ:\n'); +fprintf('• Явные методы: нужно h ~ 1/|λ| => N ~ |λ|·T шагов\n'); +fprintf(' При λ = -2000, T = 1: N ~ 2000 шагов\n\n'); + +fprintf('• Неявные методы: можно h ~ 0.01 => N ~ 100 шагов\n'); +fprintf(' Экономия: в 20 раз меньше шагов!\n\n'); + +fprintf('ИТОГ: Несмотря на бóльшую стоимость шага, неявные методы\n'); +fprintf(' БОЛЕЕ ЭФФЕКТИВНЫ для жёстких задач!\n\n'); + +fprintf('═══════════════════════════════════════════════════════════════\n'); +fprintf('5. ПРАКТИЧЕСКИЕ РЕКОМЕНДАЦИИ\n'); +fprintf('═══════════════════════════════════════════════════════════════\n\n'); + +fprintf('❶ Как определить жёсткость задачи?\n'); +fprintf(' • Наличие быстрых и медленных компонент\n'); +fprintf(' • Большие отрицательные собственные числа якобиана\n'); +fprintf(' • Отношение |λ_max/λ_min| >> 1 (stiffness ratio)\n'); +fprintf(' • Практически: явные методы становятся неустойчивыми\n\n'); + +fprintf('❷ Стратегия выбора метода:\n'); +fprintf(' • Начните с явного RK4\n'); +fprintf(' • Если наблюдается неустойчивость => задача жёсткая\n'); +fprintf(' • Переключитесь на неявные методы (ode15s в MATLAB)\n\n'); + +fprintf('❸ Для учебных целей:\n'); +fprintf(' • Нежёсткие: RK4 — отличный выбор\n'); +fprintf(' • Жёсткие: Неявная трапеция — простая реализация + хорошая устойчивость\n\n'); + +fprintf('❹ Для производственных расчётов:\n'); +fprintf(' • Используйте адаптивные методы (ode45, ode15s)\n'); +fprintf(' • Они автоматически подбирают шаг\n'); +fprintf(' • Контролируют погрешность\n\n'); + +fprintf('═══════════════════════════════════════════════════════════════\n'); +fprintf('6. ТЕОРЕТИЧЕСКОЕ ОБОСНОВАНИЕ\n'); +fprintf('═══════════════════════════════════════════════════════════════\n\n'); + +fprintf('ОБЛАСТЬ УСТОЙЧИВОСТИ (для линейного тестового уравнения y'' = λy):\n\n'); + +fprintf('• Явный Эйлер: |1 + hλ| ≤ 1\n'); +fprintf(' Круг радиуса 1 с центром в (-1, 0)\n'); +fprintf(' Устойчив только для Re(λ) < 0 и малых h\n\n'); + +fprintf('• RK4: Больше область, но всё ещё ограничена\n'); +fprintf(' Не подходит для жёстких задач\n\n'); + +fprintf('• Обратный Эйлер: |1/(1 - hλ)| ≤ 1\n'); +fprintf(' Вся левая полуплоскость Re(λ) < 0\n'); +fprintf(' БЕЗУСЛОВНО УСТОЙЧИВ!\n\n'); + +fprintf('• Неявная трапеция: |(1 + hλ/2)/(1 - hλ/2)| ≤ 1\n'); +fprintf(' Вся левая полуплоскость Re(λ) < 0\n'); +fprintf(' A-УСТОЙЧИВА + 2-й порядок точности!\n\n'); + +fprintf('• Неявный RK4 (Гаусс): A-устойчив + симплектичен\n'); +fprintf(' Идеален для гамильтоновых систем\n\n'); + +fprintf('═══════════════════════════════════════════════════════════════\n'); +fprintf('7. СРАВНЕНИЕ С АНАЛИТИЧЕСКИМ РЕШЕНИЕМ\n'); +fprintf('═══════════════════════════════════════════════════════════════\n\n'); + +fprintf('Для жёсткой задачи dy/dt = -2000y + t, y(0) = 0:\n\n'); +fprintf('Аналитическое решение:\n'); +fprintf(' y(t) = (t - 1/2000)/2000 + (1/2000²)·exp(-2000t)\n\n'); + +fprintf('Компоненты решения:\n'); +fprintf(' ① Стационарная часть: (t - 1/2000)/2000 ≈ 0.0005t\n'); +fprintf(' ② Переходная часть: 2.5×10⁻⁷·exp(-2000t)\n\n'); + +fprintf('Переходная часть затухает экспоненциально:\n'); +fprintf(' • При t = 0.005: exp(-10) ≈ 4.5×10⁻⁵\n'); +fprintf(' • При t = 0.01: exp(-20) ≈ 2×10⁻⁹\n'); +fprintf(' • Практически исчезает за t ~ 0.01\n\n'); + +fprintf('Эта БЫСТРАЯ динамика и создаёт жёсткость!\n'); +fprintf('Явные методы пытаются "отследить" быстрое затухание,\n'); +fprintf('требуя крошечного шага. Неявные методы "понимают"\n'); +fprintf('экспоненциальный характер и остаются устойчивыми.\n\n'); + +fprintf('═══════════════════════════════════════════════════════════════\n'); +fprintf(' КОНЕЦ АНАЛИЗА \n'); +fprintf('═══════════════════════════════════════════════════════════════\n'); + +%% Финальная визуализация: сравнение обеих задач + +figure('Position', [100, 100, 1400, 600]); + +% Левая панель: нежёсткая задача +subplot(1,2,1); +t_plot_4_fine = linspace(t0_4, t_end_4, 500); +plot(t_plot_4_fine, y_ref_4(t_plot_4_fine), 'k-', 'LineWidth', 2.5); +hold on; +plot(t_grid_4, y_euler_4, 'b.-', 'MarkerSize', 8); +plot(t_grid_4, y_rk4_4, 'r.-', 'MarkerSize', 8); +plot(t_grid_4, y_trap_impl_4, 'm.-', 'MarkerSize', 8); +xlabel('t', 'FontSize', 12, 'FontWeight', 'bold'); +ylabel('y(t)', 'FontSize', 12, 'FontWeight', 'bold'); +title('Нежёсткая: dy/dt = y² - t', 'FontSize', 14, 'FontWeight', 'bold'); +legend('Эталон', 'Эйлер', 'RK4', 'Неявн. трап.', 'Location', 'best'); +grid on; +hold off; + +% Правая панель: жёсткая задача +subplot(1,2,2); +t_plot_9_fine = linspace(t0_9, t_end_9, 500); +plot(t_plot_9_fine, y_ref_9(t_plot_9_fine), 'k-', 'LineWidth', 2.5); +hold on; +plot(t_grid_9, y_beuler_9, 'g.-', 'MarkerSize', 7); +plot(t_grid_9, y_trap_impl_9, 'm.-', 'MarkerSize', 7); +plot(t_grid_9, y_irk4_9, 'k.-', 'MarkerSize', 7); +xlabel('t', 'FontSize', 12, 'FontWeight', 'bold'); +ylabel('y(t)', 'FontSize', 12, 'FontWeight', 'bold'); +title('Жёсткая: dy/dt = -2000y + t', 'FontSize', 14, 'FontWeight', 'bold'); +legend('Эталон', 'Обр. Эйлер', 'Неявн. трап.', 'Неявн. RK4', 'Location', 'best'); +grid on; +hold off; + +sgtitle('Итоговое сравнение: нежёсткая vs жёсткая задача', ... + 'FontSize', 16, 'FontWeight', 'bold'); + +%% Дополнительная визуализация: области устойчивости + +figure('Position', [100, 100, 1200, 500]); + +% График слева: быстрое затухание в жёсткой задаче +subplot(1,2,1); +t_transient = linspace(0, 0.02, 200); +y_transient = y_analytical(t_transient); +y_steady = (t_transient - 1/lambda)/lambda; +plot(t_transient, y_transient, 'k-', 'LineWidth', 2, 'DisplayName', 'Полное решение'); +hold on; +plot(t_transient, y_steady, 'b--', 'LineWidth', 2, 'DisplayName', 'Стационарная часть'); +plot(t_transient, y_transient - y_steady, 'r:', 'LineWidth', 2, ... + 'DisplayName', 'Переходная часть (exp(-2000t))'); +xlabel('Время t', 'FontSize', 12, 'FontWeight', 'bold'); +ylabel('y(t)', 'FontSize', 12, 'FontWeight', 'bold'); +title('Декомпозиция решения жёсткой задачи', 'FontSize', 14, 'FontWeight', 'bold'); +legend('Location', 'best', 'FontSize', 10); +grid on; +xlim([0, 0.02]); +hold off; + +% График справа: сравнение погрешностей +subplot(1,2,2); +methods_summary = {'Эйлер', 'Явн.трап', 'RK4', 'Обр.Эйл', 'Неяв.трап', 'Неяв.RK4'}; +errors_p1_current = [err_euler_4, err_trap_expl_4, err_rk4_4, ... + err_beuler_4, err_trap_impl_4, err_irk4_4]; + +bar(1:6, log10(errors_p1_current)); +set(gca, 'XTickLabel', methods_summary, 'FontSize', 10); +xtickangle(45); +ylabel('log₁₀(Погрешность)', 'FontSize', 12, 'FontWeight', 'bold'); +title('Сравнение погрешностей (нежёсткая задача)', 'FontSize', 14, 'FontWeight', 'bold'); +grid on; +ylim([-8, 0]); + +sgtitle('Детальный анализ', 'FontSize', 16, 'FontWeight', 'bold'); + +fprintf('\n✓ Все графики построены!\n'); +fprintf('✓ Анализ завершён успешно!\n\n'); \ No newline at end of file diff --git a/labs/lab1/code/main.py b/labs/lab1/code/main.py new file mode 100644 index 0000000..f521f94 --- /dev/null +++ b/labs/lab1/code/main.py @@ -0,0 +1,189 @@ +import matplotlib.pyplot as plt +import math +import numpy as np +from scipy.integrate import solve_ivp +from scipy.optimize import fsolve + + +class ODESolver: + def euler_method(self, f, t_span, y0, h): + t0, t_end = t_span + N = int((t_end - t0) / h) + t = np.linspace(t0, t_end, N + 1) + y = np.zeros(N + 1) + y[0] = y0 + + for i in range(N): + y[i + 1] = y[i] + h * f(t[i], y[i]) + + return t, y + + def explicit_trapezoid(self, f, t_span, y0, h): + t0, t_end = t_span + N = int((t_end - t0) / h) + t = np.linspace(t0, t_end, N + 1) + y = np.zeros(N + 1) + y[0] = y0 + + for i in range(N): + k1 = f(t[i], y[i]) + k2 = f(t[i] + h, y[i] + h * k1) + y[i + 1] = y[i] + (h / 2) * (k1 + k2) + + return t, y + + def rk4_method(self, f, t_span, y0, h): + t0, t_end = t_span + N = int((t_end - t0) / h) + t = np.linspace(t0, t_end, N + 1) + y = np.zeros(N + 1) + y[0] = y0 + + for i in range(N): + k1 = f(t[i], y[i]) + k2 = f(t[i] + h/2, y[i] + h/2 * k1) + k3 = f(t[i] + h/2, y[i] + h/2 * k2) + k4 = f(t[i] + h, y[i] + h * k3) + y[i + 1] = y[i] + h/6 * (k1 + 2*k2 + 2*k3 + k4) + + return t, y + + def backward_euler(self, f, t_span, y0, h): + t0, t_end = t_span + N = int((t_end - t0) / h) + t = np.linspace(t0, t_end, N + 1) + y = np.zeros(N + 1) + y[0] = y0 + + for i in range(N): + t_next = t[i + 1] + + def equation(z): + return z - h * f(t_next, z) - y[i] + + y[i + 1] = fsolve(equation, y[i])[0] + + return t, y + + def implicit_trapezoid(self, f, t_span, y0, h): + t0, t_end = t_span + N = int((t_end - t0) / h) + t = np.linspace(t0, t_end, N + 1) + y = np.zeros(N + 1) + y[0] = y0 + + for i in range(N): + t_n = t[i] + t_np1 = t[i + 1] + y_n = y[i] + + def equation(z): + return z - y_n - (h/2) * (f(t_n, y_n) + f(t_np1, z)) + + y[i + 1] = fsolve(equation, y_n)[0] + + return t, y + +def problem_1_4(): + print("=== ЗАДАЧА 1.4: y' = y² - t, y(0) = 1 ===") + + def f(t, y): + return y**2 - t + + t_span = [0, 1.0] + y0 = 1 + h = 0.25 + + solver = ODESolver() + + t_euler, y_euler = solver.euler_method(f, t_span, y0, h) + t_trap, y_trap = solver.explicit_trapezoid(f, t_span, y0, h) + t_rk4, y_rk4 = solver.rk4_method(f, t_span, y0, h) + + sol_ref = solve_ivp(f, t_span, [y0], method='RK45', + rtol=1e-10, atol=1e-12, dense_output=True) + t_ref = np.linspace(t_span[0], t_span[1], 100) + y_ref = sol_ref.sol(t_ref)[0] + + y_ref_euler = sol_ref.sol(t_euler)[0] + y_ref_trap = sol_ref.sol(t_trap)[0] + y_ref_rk4 = sol_ref.sol(t_rk4)[0] + + err_euler = np.linalg.norm(y_euler - y_ref_euler) + err_trap = np.linalg.norm(y_trap - y_ref_trap) + err_rk4 = np.linalg.norm(y_rk4 - y_ref_rk4) + + print(f"Ошибка метода Эйлера: {err_euler:.2e}") + print(f"Ошибка явного метода трапеций: {err_trap:.2e}") + print(f"Ошибка метода РК4: {err_rk4:.2e}") + + plt.figure(figsize=(12, 8)) + plt.plot(t_ref, y_ref, 'k-', linewidth=5, label='Эталонное решение (solve_ivp)') + plt.plot(t_euler, y_euler, 'bo-', markersize=10, label='Метод Эйлера') + plt.plot(t_trap, y_trap, 'c*-', markersize=15, label='Явный метод трапеций') + plt.plot(t_rk4, y_rk4, 'rs-', markersize=5, label='Метод РК4') + + plt.xlabel('t') + plt.ylabel('y(t)') + plt.title('Задача 1.4: y\' = y² - t, y(0) = 1') + plt.legend() + plt.grid(True, alpha=0.3) + plt.show() + + return t_euler, y_euler, t_trap, y_trap, t_rk4, y_rk4, t_ref, y_ref + + +if __name__ == "__main__": + result1 = problem_1_4() + +def problem_2_4(): + print("\n=== ЗАДАЧА 2.4: y' = -2000y + t, y(0) = 0 ===") + + def f(t, y): + return -2000 * y + t + + t_span = [0, 1] + y0 = 0 + h = 1 + + solver = ODESolver() + + t_euler, y_euler = solver.euler_method(f, t_span, y0, h) + t_beuler, y_beuler = solver.backward_euler(f, t_span, y0, h) + t_itrap, y_itrap = solver.implicit_trapezoid(f, t_span, y0, h) + + sol_ref = solve_ivp(f, t_span, [y0], method='Radau', + rtol=1e-10, atol=1e-12, dense_output=True) + t_ref = np.linspace(t_span[0], t_span[1], 100) + y_ref = sol_ref.sol(t_ref)[0] + + y_ref_euler = sol_ref.sol(t_euler)[0] + y_ref_beuler = sol_ref.sol(t_beuler)[0] + y_ref_itrap = sol_ref.sol(t_itrap)[0] + + err_euler = np.linalg.norm(y_euler - y_ref_euler) + err_beuler = np.linalg.norm(y_beuler - y_ref_beuler) + err_itrap = np.linalg.norm(y_itrap - y_ref_itrap) + + print(f"Ошибка явного метода Эйлера: {err_euler:.2e}") + print(f"Ошибка неявного метода Эйлера: {err_beuler:.2e}") + print(f"Ошибка неявного метода трапеций: {err_itrap:.2e}") + + plt.figure(figsize=(12, 8)) + plt.plot(t_ref, y_ref, 'k-', linewidth=5, label='Эталонное решение (solve_ivp)') + plt.plot(t_euler, y_euler, 'bo-', markersize=4, label='Явный метод Эйлера') + plt.plot(t_beuler, y_beuler, 'g^-', markersize=4, label='Неявный метод Эйлера') + plt.plot(t_itrap, y_itrap, 'm*-',linewidth=2.2, markersize=4, label='Неявный метод трапеций') + + plt.xlabel('t') + plt.ylabel('y(t)') + plt.title('Задача 2.4: y\' = -2000y + t, y(0) = 0 (жесткая система)') + plt.legend() + plt.grid(True, alpha=0.3) + plt.show() + + return t_euler, y_euler, t_beuler, y_beuler, t_itrap, y_itrap, t_ref, y_ref + + +if __name__ == "__main__": + result2 = 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