* Load is given in academic hour (1 academic hour = 45 minutes)
To introduce students to modern methods in numerical analysis, in the area of ordinary (ODE) and partial differential equations (PDE), with an emphasis on their practical solution on computers.
Final difference method for boundary value problem for ODE. Partial differential
equations, final difference method for Poisson equation, Liebmann method. Final
element method (FEM) in one dimension and for elliptic (PDE) boundary value problems, variational formulation, Ritz-Galerkin method, basis functions and form functions, spaces of final elements. Approximation of domain, local coordinates and the nesting algorithm, the knot numeration problem. Methods for parabolic equations. Convergence of FEM. Classification of second order PDE in two dimensions. Hyperbolic equations of the first and second order, method of characteristics, propagation of discontinuities, Lax-Wendroff formulae and Courant-Friedrichs convergence condition.
After the successful completion of the subject Numerical methods in physics, the student will be able to:
1. express the basic definition and theorems associated with the ordinary and partial differential equations, as well as with the approximation methods;
2. differentiate the methods for solving initial and boundary value problems for ordinary and partial differential equations;
3. choose and apply the correct approximation methods for the given problem;
4. derive an analogous approximation method with certain properties;
5. analyze a given approximation method;
6. write a simple computer program for solving a given problem.
Following lectures, study of notes and literature, analysis of examples and practicing, analysis of methods and practicing, analysis of computer programs and the results obtained by solving problems on the computer and practicing.
Lectures; solving examples; analysis of the methods; presentation of the
computer programs and their results.
METHODS OF MONITORING AND VERIFICATION:
Written exam through midterm exams; writing and presenting programming assignments; oral exam.
TERMS FOR RECEIVING THE SIGNATURE:
Regular attendance to the lectures, and achievement of minimal 17 points out of 56 on
1. Two mid-term exams, 28 points each (together 56 points)
2. One programming assignment, 24 points
3. Final exam, 20 points
1. During the semester, students write two mid-term exams. Mid-term exams include also some theoretical questions.
2. Minimal condition for passing the exam is achievement of 17 points.
3. For students who were not able to achieve the minimal number of points, one makeup mid-term exam will be organized, which includes material of the whole semester. Maximum number of points on the makeup exam is 56. Minimal condition for passing this exam is achievement of 17 points. For students who approach the makeup mid-term exam, the points from the regular mid-term exams are reset.
1. During the semester, one individual programing assignment is set, which must be solved within the time limit, which will be announced on the web page of the course. Each assignment, in principle, includes a solution implemented in F90/F95, and is explained to the lecturer.
2. Minimal condition for passing is achievement of 10 points.
1. Final exam consists of an oral exam in front of the lecturer, which includes the material of the whole course, and may include some tasks and test of the practical knowledge on the computer.
2. The students who have passed the mid-term exams and the programming assignment may approach the final exam.
Minimal number of points for passing grade is 45. The final grade is determined by the
- COMPULSORY LITERATURE:
Bellman, R.E., R.E. Kalaba: Quasilinearization and Nonlinear Boundary-Value Problems, Elsevier N.Y. 1965.
Strang, G., G. J. Fix: An Analysis of the FEM, Prentice-Hall, 1973.
Press, W. H., B.P. Flannery, S. A. Teukolsky, W. T. Vetterling: Numerical Recipes, Cambridge univ. press, 1987.
Smith, G.D.: Numerical Solution of PDE: Finite Difference Methods, Clarendon press, Oxford, 1978.
- Strang, G., G. J. Fix: An Analysis of the FEM, Prentice-Hall, 1973.
- Press, W. H., B.P. Flannery, S. A. Teukolsky, W. T. Vetterling: Numerical Recipes, Cambridge univ. press, 1987.
- Smith, G.D.: Numerical Solution of PDE: Finite Difference Methods, Clarendon press, Oxford, 1978.
Numerical Methods in Physics 1