ECTS - Finite Difference Methods for PDEs
Finite Difference Methods for PDEs (MATH524) Course Detail
| Course Name | Course Code | Season | Lecture Hours | Application Hours | Lab Hours | Credit | ECTS |
|---|---|---|---|---|---|---|---|
| Finite Difference Methods for PDEs | MATH524 | 3 | 0 | 0 | 3 | 5 |
| Pre-requisite Course(s) |
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| Consent of the department |
| Course Language | English |
|---|---|
| Course Type | N/A |
| Course Level | Natural & Applied Sciences Master's Degree |
| Mode of Delivery | Face To Face |
| Learning and Teaching Strategies | Lecture, Question and Answer, Problem Solving. |
| Course Lecturer(s) |
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| Course Objectives | This graduate course is designed to give students in applied mathematics the expertise necessary to understand, construct and use finite difference methods for the numerical solution of partial differential equations. The emphasis is on implementation of various finite difference schemes to some model partial differential equations, finding numerical solutions, evaluating numerical results and understands how and why results might be good or bad based on consistency, stability and convergence of finite difference scheme. |
| Course Learning Outcomes |
The students who succeeded in this course;
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| Course Content | Finite difference method, parabolic equations: explicit and implicit methods, Richardson, Dufort-Frankel and Crank-Nicolson schemes; hyperbolic equations: Lax-Wendroff, Crank-Nicolson, box and leap-frog schemes; elliptic equations: consistency, stability and convergence of finite different methods for numerical solutions of partial differential equ |
Weekly Subjects and Releated Preparation Studies
| Week | Subjects | Preparation |
|---|---|---|
| 1 | -Classification of Partial Differential Equations (PDE): Parabolic, Hyperbolic and Elliptic PDE. -Boundary conditions. -Finite difference methods, Finite difference operators | [Lapidus] p:1-3, 12, 13, 28-30, 34-41, [Smith] p.1-8 |
| 2 | Parabolic equations: -Explicit methods -Truncation error, consistence, order of accuracy | [Morton & Mayers] p.10-16 |
| 3 | -Convergence of the explicit schemes. -Stability by Fourier analysis and matrix method | [Morton & Mayers] p.16-22 [Smith] p.60-64 |
| 4 | -Implicit methods. -Thomas algorithm -Richardson scheme | [Morton & Mayers] p.22-26,38, 39 |
| 5 | -Duforth-Frankel’s explicit scheme -Boundary conditions, | [Smith] p.32-40,94 [Morton & Mayers] p. 39-42 |
| 6 | -Crank-Nicolson implicit scheme and its stability -Iterative methods for solving implicit scheme s | [Smith].p.17-20, 64-67, 24-32, |
| 7 | -Finite difference methods for variable coefficient PDE. | [Morton & Mayers] p.46-51,54-56 |
| 8 | Hyperbolic equations: -The upwind scheme and its local truncation error, stabilirty and convergence. -The Courant, Friedrichs and Lewy (CLF) condition. | [Morton & Mayers] p:89-95 |
| 9 | -The Lax-Wendroff scheme and its stability -The Crank-Nicolson scheme and its stability | [Morton & Mayers] p.100, [ Strikwerda] p.63, 77 |
| 10 | Midterm Exam | |
| 11 | -The box scheme and its order of accuracy -The Leap-frog scheme and its stability | [Morton & Mayers] p.116-118, 123,124 |
| 12 | Elliptic equations: -A model problem:Poisson equation -Boundary conditions on a curve boundary | [Morton & Mayers] p.194,195, 199-203 |
| 13 | -Basic iterative schemes | [Morton & Mayers] p.237-244 |
| 14 | -Alternating Direction Implicit method | [Smith] p.151-153 |
| 15 | Review | |
| 16 | Final Exam |
Sources
| Course Book | 1. K.W. Morton, D.F. Mayers, Numerical Solutions of Partial Differential Equations, 2nd Edition, Cambridge University Press, 2005. |
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| Other Sources | 2. G.D. Smith, Numerical Solutions of Partial Differential Equations, Oxford University Press, 1969 |
| 3. L. Lapidus, G.F. Pinder, Numerical Solutions of Partial Differential Equations in Science and Engineering, John Wiley & Sons, Inc. 1999. | |
| 4. J.C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, 2nd Edition, SIAM, 2004 |
Evaluation System
| Requirements | Number | Percentage of Grade |
|---|---|---|
| Attendance/Participation | - | - |
| Laboratory | - | - |
| Application | - | - |
| Field Work | - | - |
| Special Course Internship | - | - |
| Quizzes/Studio Critics | - | - |
| Homework Assignments | 4 | 20 |
| Presentation | 1 | 10 |
| Project | 1 | 10 |
| Report | - | - |
| Seminar | - | - |
| Midterms Exams/Midterms Jury | 1 | 30 |
| Final Exam/Final Jury | 1 | 30 |
| Toplam | 8 | 100 |
| Percentage of Semester Work | 70 |
|---|---|
| Percentage of Final Work | 30 |
| Total | 100 |
Course Category
| Core Courses | |
|---|---|
| Major Area Courses | |
| Supportive Courses | X |
| Media and Managment Skills Courses | |
| Transferable Skill Courses |
The Relation Between Course Learning Competencies and Program Qualifications
| # | Program Qualifications / Competencies | Level of Contribution | ||||
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | ||
| 1 | Has the ability to apply scientific knowledge gained in the undergraduate education and to expand and extend knowledge in the same or in a different area. | X | ||||
| 2 | Has the ability to obtain, to evaluate, to interpret and to apply information by doing scientific research. | X | ||||
| 3 | Can apply gained knowledge and problem solving abilities in inter-disciplinary research. | X | ||||
| 4 | Has the ability to work independently within research area, to state the problem, to develop solution techniques, to solve the problem, to evaluate the obtained results and to apply them when necessary. | X | ||||
| 5 | Takes responsibility individually and as a team member to improve systematic approaches to produce solutions in unexpected complicated situations related to the area of study. | X | ||||
| 6 | Can develop strategies, implement plans and principles on the area of study and can evaluate obtained results within the framework. | X | ||||
| 7 | Can develop and extend the knowledge in the area and to use them with scientific, social and ethical responsibility. | X | ||||
| 8 | Has the ability to follow recent developments within the area of research, to support research with scientific arguments and data, to communicate the information on the area of expertise in a systematically by means of written report and oral/visual presentation. | X | ||||
| 9 | Has proficiency in English language and has the ability to communicate with colleagues and to follow the innovations in mathematics and related fields. | X | ||||
| 10 | Has software and hardware knowledge in the area of expertise, and has proficient information and communication technology knowledge. | X | ||||
| 11 | Has professional ethical consciousness and responsibility which takes into account the universal and social dimensions in the process of data collection, interpretation, implementation and declaration of results in mathematics and its applications. | X | ||||
ECTS/Workload Table
| Activities | Number | Duration (Hours) | Total Workload |
|---|---|---|---|
| Course Hours (Including Exam Week: 16 x Total Hours) | 16 | 3 | 48 |
| Laboratory | |||
| Application | |||
| Special Course Internship | |||
| Field Work | |||
| Study Hours Out of Class | 14 | 2 | 28 |
| Presentation/Seminar Prepration | 1 | 8 | 8 |
| Project | 1 | 7 | 7 |
| Report | |||
| Homework Assignments | 4 | 3 | 12 |
| Quizzes/Studio Critics | |||
| Prepration of Midterm Exams/Midterm Jury | 1 | 10 | 10 |
| Prepration of Final Exams/Final Jury | 1 | 12 | 12 |
| Total Workload | 125 | ||