title: ÐÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ðµ Ð¼ÐµÑÐ¾Ð´Ð¾Ð² ÑÐµÑÐµÐ½Ð¸Ñ Ð¶ÑÑÑÐºÐ¸Ñ ÑÐ¸ÑÑÐµÐ¼ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ð¹ Ð½Ð° Ð¿ÑÐ¸Ð¼ÐµÑÐµ Ð·Ð°Ð´Ð°ÑÐ¸ Ð·Ð°Ð¿Ð¾Ð»Ð½ÐµÐ½Ð¸Ñ ÑÑÐ´Ð¾Ð¿ÑÐ¾Ð¿ÑÑÐºÐ½Ð¾Ð³Ð¾ ÑÐ»ÑÐ·Ð°
title: Ð²ÑÐ¿ÑÑÐºÐ½Ð°Ñ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð±Ð°ÐºÐ°Ð»Ð°Ð²ÑÐ°
titleType: Subtitle
title: Study of methods for solving rigid systems of equations by the example of the ship lock filling problem
Creators: name: ÐÑÐ¾ÑÐ¸ÐºÐ¾Ð², Ð¡ÐµÑÐ³ÐµÐ¹
Publisher: name: Ð¡Ð°Ð½ÐºÑ-ÐÐµÑÐµÑÐ±ÑÑÐ³ÑÐºÐ¸Ð¹ Ð¿Ð¾Ð»Ð¸ÑÐµÑÐ½Ð¸ÑÐµÑÐºÐ¸Ð¹ ÑÐ½Ð¸Ð²ÐµÑÑÐ¸ÑÐµÑ ÐÐµÑÑÐ° ÐÐµÐ»Ð¸ÐºÐ¾Ð³Ð¾
Year: 2023
Subjects: Ð´Ð¸ÑÑÐµÑÐµÐ½ÑÐ¸Ð°Ð»ÑÐ½Ð¾-Ð°Ð»Ð³ÐµÐ±ÑÐ°Ð¸ÑÐµÑÐºÐ¸Ðµ Ð·Ð°Ð´Ð°ÑÐ¸
Ð½ÐµÑÐ²Ð½ÑÐµ Ð¼ÐµÑÐ¾Ð´Ñ
Ð¶ÐµÑÑÐºÐ¸Ðµ Ð·Ð°Ð´Ð°ÑÐ¸
Ð¼ÐµÑÐ¾Ð´ ÑÑÐ½Ð³Ðµ-ÐºÑÑÑÑ
Ð¼ÐµÑÐ¾Ð´ Ð°Ð´Ð°Ð¼ÑÐ°
Ð¼ÐµÑÐ¾Ð´ ÑÐ¹Ð»ÐµÑÐ°
Ð¼Ð¾Ð´Ð¸ÑÐ¸ÑÐ¸ÑÐ¾Ð²Ð°Ð½Ð½ÑÐ¹ Ð¼ÐµÑÐ¾Ð´ ÑÐ¹Ð»ÐµÑÐ°
Ð²ÑÑÐ¸ÑÐ»Ð¸ÑÐµÐ»ÑÐ½Ð°Ñ ÑÐ¾ÑÐ½Ð¾ÑÑÑ
differential-algebraic problems
implicit methods
stiff problems
runge-kutta method
adams method
euler method
modified euler method
computational accuracy

description: Ð Ð´Ð°Ð½Ð½Ð¾Ð¹ ÑÐ°Ð±Ð¾ÑÐµ Ð¸Ð·Ð»Ð¾Ð¶ÐµÐ½Ð¾ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ðµ ÑÐµÑÐµÐ½Ð¸Ñ Ð¶ÐµÑÑÐ¸Ñ ÑÐ¸ÑÑÐµÐ¼ Ð¸ Ð´Ð¸ÑÑÐµÑÐµÐ½ÑÐ¸Ð°Ð»ÑÐ½Ð¾- Ð°Ð»Ð³ÐµÐ±ÑÐ°Ð¸ÑÐµÑÐºÐ¸Ñ ÑÐ¸ÑÑÐµÐ¼ ÑÐ²Ð½ÑÐ¼Ð¸ Ð¸ Ð½ÐµÑÐ²Ð½ÑÐ¼Ð¸ Ð¼ÐµÑÐ¾Ð´Ð°Ð¼Ð¸. Ð¡Ð¸ÑÑÐµÐ¼Ñ Ð±ÑÐ»Ð¸ Ð¿Ð¾ÑÑÑÐ¾ÐµÐ½Ñ Ð½Ð° Ð¾ÑÐ½Ð¾Ð²Ðµ Ð¼Ð¾Ð´ÐµÐ»ÑÐ½Ð¾Ð¹ Ð·Ð°Ð´Ð°ÑÐ¸ ÑÐ°Ð±Ð¾ÑÑ ÑÑÐ´Ð¾Ð¿ÑÐ¾Ð¿ÑÑÐºÐ½Ð¾Ð³Ð¾ ÑÐ»ÑÐ·Ð°. Ð ÑÐ¾Ð´Ðµ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ: 1. ÐÑÐ»Ð¸ ÑÐ°ÑÑÐ¼Ð¾ÑÑÐµÐ½Ñ ÑÐ²Ð½ÑÐµ - Ð¼ÐµÑÐ¾Ð´ Ð­Ð¹Ð»ÐµÑÐ°, Ð¼Ð¾Ð´Ð¸ÑÐ¸ÑÐ¸ÑÐ¾Ð²Ð°Ð½Ð½ÑÐ¹ Ð¼ÐµÑÐ¾Ð´ Ð­Ð¹Ð»ÐµÑÐ°, Ð¼ÐµÑÐ¾Ð´ Ð ÑÐ½Ð³Ðµ-ÐÑÑÑÑ 3-Ð³Ð¾ Ð¿Ð¾ÑÑÐ´ÐºÐ° Ñ ÐºÐ¾ÑÑÑÐ¸ÑÐ¸ÐµÐ½ÑÐ¾Ð¼ 1/2, Ð¼ÐµÑÐ¾Ð´ ÐÐ´Ð°Ð¼ÑÐ° 3-Ð³Ð¾ Ð¿Ð¾ÑÑÐ´ÐºÐ°; Ð¸ Ð½ÐµÑÐ²Ð½ÑÐµ Ð¼ÐµÑÐ¾Ð´Ñ - Ð½ÐµÑÐ²Ð½ÑÐ¹ Ð¼ÐµÑÐ¾Ð´ Ð­Ð¹Ð»ÐµÑÐ°. 2. ÐÑÐ»Ð¸ ÑÐµÐ°Ð»Ð¸Ð·Ð¾Ð²Ð°Ð½Ñ Ð½Ð° ÑÐ·ÑÐºÐµ C++ ÑÐ¾Ð¾ÑÐ²ÐµÑÑÑÐ²ÑÑÑÐ¸Ðµ Ð¼ÐµÑÐ¾Ð´Ñ. 3. ÐÐµÑÐ¾Ð´Ð°Ð¼Ð¸ Ð±ÑÐ»Ð¸ ÑÐµÑÐµÐ½Ñ ÑÐµÑÑÐ¾Ð²Ð°Ñ Ð¸ Ð¿Ð¾Ð»Ð½Ð°Ñ Ð·Ð°Ð´Ð°ÑÐ¸ Ð´Ð»Ñ Ð¿ÑÐ¾Ð²ÐµÑÐºÐ¸ Ð¸Ñ ÐºÐ¾ÑÑÐµÐºÑÐ½Ð¾Ð¹ ÑÐ°Ð±Ð¾ÑÑ Ð¸ Ð²ÐµÑÐ¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¸ Ð¼ÐµÑÐ¾Ð´Ð¾Ð² Ð´Ð»Ñ Ð·Ð°Ð´Ð°Ñ Ð½Ðµ Ð¸Ð¼ÐµÑÑÐ¸Ñ Ð°Ð½Ð°Ð»Ð¸ÑÐ¸ÑÐµÑÐºÐ¾Ð³Ð¾ ÑÐµÑÐµÐ½Ð¸Ñ. 4. ÐÑÐ¾Ð¸Ð·Ð²ÐµÐ´ÐµÐ½Ð¾ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ðµ Ð¼ÐµÑÐ¾Ð´Ð¾Ð² Ð¿Ð¾ ÑÐ¾ÑÐ½Ð¾ÑÑÐ¸ ÑÐµÑÐµÐ½Ð¸Ñ Ð¾Ñ ÑÐ°Ð³Ð° ÑÐµÑÐºÐ¸. 5. ÐÑÑÐ²Ð»ÐµÐ½Ñ Ð½ÐµÐ´Ð¾ÑÑÐ°ÑÐºÐ¸ Ð¸ Ð¿ÑÐµÐ¸Ð¼ÑÑÐµÑÑÐ²Ð° Ð¼ÐµÑÐ¾Ð´Ð¾Ð² Ð¿ÑÐ¸ ÑÐµÑÐµÐ½Ð¸Ð¸ Ð¿Ð¾Ð»Ð½Ð¾Ð¹ Ð·Ð°Ð´Ð°ÑÐ¸. 6. Ð¡Ð´ÐµÐ»Ð°Ð½Ñ Ð²ÑÐ²Ð¾Ð´Ñ Ð¸ ÑÐµÐºÐ¾Ð¼ÐµÐ½Ð´Ð°ÑÐ¸Ð¸ Ð¿Ð¾ Ð¿ÑÐ¸Ð¼ÐµÐ½ÐµÐ½Ð¸Ñ ÑÐ²Ð½ÑÑ Ð¸ Ð½ÐµÑÐ²Ð½ÑÑ Ð¼ÐµÑÐ¾Ð´Ð¾Ð² Ðº ÑÐµÑÐµÐ½Ð¸Ñ Ð´Ð°Ð½Ð½Ð¾Ð³Ð¾ ÐºÐ»Ð°ÑÑÐ° Ð·Ð°Ð´Ð°Ñ ÐÐ»Ñ ÑÐµÐ°Ð»Ð¸Ð·Ð°ÑÐ¸Ð¸ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ Ð¸ÑÐ¿Ð¾Ð»ÑÐ·Ð¾Ð²Ð°Ð»ÑÑ ÑÐ·ÑÐº C++ Ð¸ Ð¿Ð°ÐºÐµÑ Ð¿ÑÐ¸ÐºÐ»Ð°Ð´Ð½ÑÑ Ð¿ÑÐ¾Ð³ÑÐ°Ð¼Ð¼ MATLAB. Ð ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÐµ, Ð±ÑÐ»Ð¾ Ð¿Ð¾ÐºÐ°Ð·Ð°Ð½Ð¾, ÑÑÐ¾ Ð´Ð»Ñ Ð¿Ð¾Ð»Ð½Ð¾Ð¹ Ð´Ð¸ÑÑÐµÑÐµÐ½ÑÐ¸Ð°Ð»ÑÐ½Ð¾- Ð°Ð»Ð³ÐµÐ±ÑÐ°Ð¸ÑÐµÑÐºÐ¾Ð¹ Ð·Ð°Ð´Ð°ÑÐ¸ ÑÐ²Ð½ÑÐµ Ð¸ Ð½ÐµÑÐ²Ð½ÑÐµ Ð¼ÐµÑÐ¾Ð´Ñ ÑÐ°Ð±Ð¾ÑÐ°ÑÑ Ð¾Ð´Ð¸Ð½Ð°ÐºÐ¾Ð²Ð¾. Ð¡ Ð´ÑÑÐ³Ð¾Ð¹ ÑÑÐ¾ÑÐ¾Ð½Ñ Ð´Ð»Ñ Ð¶ÐµÑÑÐ¸Ñ Ð·Ð°Ð´Ð°Ñ ÑÐ²Ð½ÑÐµ Ð¼ÐµÑÐ¾Ð´Ñ ÑÐ°Ð±Ð¾ÑÐ°ÑÑ Ð½Ðµ ÐºÐ¾ÑÑÐµÐºÑÐ½Ð¾, Ð° Ð½ÐµÑÐ²Ð½ÑÐµ Ð¿Ð¾ÐºÐ°Ð·ÑÐ²Ð°Ð»Ð¸ ÑÐ¾ÑÐ¾ÑÐ¸Ðµ ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÑ Ð¿Ð¾ ÑÑÐ¾Ð´Ð¸Ð¼Ð¾ÑÑÐ¸ Ðº ÑÐ¾ÑÐ½Ð¾Ð¼Ñ ÑÐµÑÐµÐ½Ð¸Ñ. Ð ÐµÐ·ÑÐ»ÑÑÐ°ÑÑ Ð´Ð°Ð½Ð½Ð¾Ð¹ ÑÐ°Ð±Ð¾ÑÑ Ð¼Ð¾Ð³ÑÑ Ð±ÑÑÑ Ð¸ÑÐ¿Ð¾Ð»ÑÐ·Ð¾Ð²Ð°Ð½Ñ Ð´Ð»Ñ Ð²ÑÐ±Ð¾ÑÐ° Ð¼ÐµÑÐ¾Ð´Ð° ÑÐµÑÐµÐ½Ð¸ÑÐ·Ð°Ð´Ð°ÑÐ¸ Ð¸Ð· ÐºÐ»Ð°ÑÑÐ° Ð´Ð¸ÑÑÐµÑÐµÐ½ÑÐ¸Ð°Ð»ÑÐ½Ð¾-Ð°Ð»Ð³ÐµÐ±ÑÐ°Ð¸ÑÐµÑÐºÐ¸Ñ Ð¸ Ð¶ÐµÑÑÐºÐ¸Ñ Ð·Ð°Ð´Ð°Ñ.
description: This paper presents a study of the solution of rigid systems and differential algebraic systems by explicit and implicit methods. The systems were constructed on the basis of a model problem for the operation of a ship-gateway. In the course of the research: 1. The explicit methods - Euler method, modified Euler method, 3rd order Runge-Kutta method with coefficient 1/2, 3rd order Adams method; and implicit methods - implicit Euler method were considered. 2. The corresponding methods were implemented in C++. 3. Test and complete problems were solved to check their correct work and to verify the methods for the problems that do not have analytical solution. 4. The methods were compared in terms of accuracy of the solution by the grid step. 5. Disadvantages and advantages of the methods in solving the full problem are revealed. 6. Conclusions and recommendations on the application of explicit and implicit methods to the solution of the given type of problems have been made. The language C++ and MATLAB package of applied programs were used to implement the study. As a result, it was shown that for the full differential-algebraic problem, explicit and implicit methods work equally. On the other hand, for rigid problems the explicit methods do not work correctly, while the implicit methods showed good results in terms of convergence to the exact solution. The results of this work can be used to choose a method for solving the problem from the class of differential-algebraic and rigid problems.