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Numerical methods of mathematical physics

ННД.28

Обов’язкова дисципліна для ОП

First

2022/2023

7 Semester

4

LO 3. Formalize tasks formulated in the language of a specific subject area; formulate their mathematical statement and choose a rational solution method; to solve the obtained problems by analytical and numerical methods, to evaluate the accuracy and reliability of the obtained results. LO 5. Be able to develop and use in practice algorithms related to approximation of functional dependencies, numerical differentiation and integration, solution of systems of algebraic, differential and integral equations, solution of boundary value problems, search for optimal solutions.

Full-time form

As a result of studying the academic discipline, students should: Know: projection-variational methods of solving boundary value problems, methods of construction, research and implementation of difference methods and the method of finite elements for boundary value problems for ordinary differential equations and partial differential equations. Be able: apply methods of construction, research and implementation of numerical methods for solving mathematical models based on the application of the main types of mathematical physics problems; be able to apply projection-variational methods, difference methods and the finite element method for solving boundary value problems for ordinary differential equations and partial differential equations.

1 Topic 1. Projection-variational methods of solving operator equations. Methods of moments, Bubnov-Galyorkin, collocation, Ritz, least squares. Formulation of the problem in generalized spaces. Convergence of methods. Examples of application of methods to boundary value problems. 2 Topic 2. Basic provisions of the grid method. The concept of a difference scheme. Approximation, stability, convergence of difference schemes. Approximation methods of basic differential operators. The main methods of construction of difference problems: the method of BZDPR, undefined coefficients, the integro-interpolation method and its modifications, methods of approximation of the quadratic functional and integral identity. 3 Topic 3. Approximation of boundary conditions of the third kind. Schemes of a higher order of approximation. Convergence of difference schemes. Monotonic difference schemes. 4 Topic 4. Finite element method. The algorithm of the method and its convergence. Construction of basic functions. 5 Topic 5. Difference schemes for non-stationary problems. Schemes with weighting factors for the heat conduction equation and the string vibration equation. Stability of two-layer and three-layer schemes. 6 Topic 6. Economic difference schemes for multidimensional problems. Method of variable directions, locally one-dimensional schemes.

4. Makarov V.L., Gavriliuk I.P. Metodi obchislen-, t.2,Kiїv,Vishcha shkola,1995. ..

Lectures, laboratory classes, independent work

Semester assessment: Maximum number: 60/36 points: 1. Control work: PH 1.1, PH 1.2, PH 1.3, PH 1.4 – 20/12 points. 2. Laboratory work No. 1: RN 1.1, RN 2.1, RN 3.1, RN 3.2, RN 4.1, RN 4.2 – 10/6 points. 3. Laboratory work No. 2: RN 1.2, RN 2.2, RN 3.1, RN 3.2, RN 4.1, RN 4.2 – 8/5 points. 4. Laboratory work No. 3: RN 1.3, RN 2.3, RN 3.1, RN 3.2, RN 4.1, RN 4.2 – 7/4 points. 5. Laboratory work No. 4: PH 1.4, PH 2.4, PH 3.1, PH 3.2, PH 4.1, PH 4.2 – 10/4 points. 6. Assessment of independent work: PH 1.1, PH 1.2, PH 1.3, PH 1.4, PH2.1, PH 2.2, PH 2.3, PH 2.4, PH 4.1, PH 4.2 - 5/3 points. Final evaluation (in the form of an exam): 1. The maximum number of points that can be obtained by a student: 40/24 points/(s). 2. Learning outcomes that will be evaluated: PH 1.1, PH 1.2, PH 1.3, PH 1.4, PH 2.1, PH 2.2, PH 2.3, PH 2.4. 3. Form of conduct: written work. 4. Types of tasks: 4 written tasks (1 theoretical question and 3 practical tasks).

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