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 | Area Elective | 3 | 0 | 0 | 3 | 5 |
| Pre-requisite Course(s) |
|---|
| N/A |
| Course Language | English |
|---|---|
| Course Type | Technical Electives (Group 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) |
|
| 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. |
|---|---|
| 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 | X |
| Supportive Courses | |
| 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 | Gains accumulated knowledge on mathematics, science and mechatronics engineering; develops an ability to apply the theoretical and applied knowledge of mathematics, science and mechatronics engineering to model and analyze mechatronics engineering problems. | X | ||||
| 2 | Develops ability to differentiate, identify, formulate, and solve complex engineering problems; develops ability to select and implement proper analysis, modeling and implementation techniques for the identified engineering problems. | X | ||||
| 3 | Develops ability to design a complex system, product, component or process to meet the requirements under realistic constraints and conditions; develops ability to apply contemporary design methodologies; an ability to implement effective engineering creativity techniques in mechatronics engineering. (Realistic constraints and conditions includes economics, environment, sustainability, producibility, ethics, human health, social and political problems.) | X | ||||
| 4 | Gains ability to develop, select and use modern techniques, skills and tools for application of mechatronics engineering and robot technologies; develops ability to use information and communications technologies effectively. | |||||
| 5 | Develops ability to design experiments, perform experiments, collect and analyze data and assess the results for investigated problems on mechatronics engineering and robot technologies. | |||||
| 6 | Develops ability to work effectively on single disciplinary and multi-disciplinary teams; gains ability for individual work; develops ability to communicate and collaborate/cooperate effectively with other disciplines and scientific/engineering domains or working areas, ability to work with other disciplines. | |||||
| 7 | Develops ability to express creative and original concepts and ideas orally or written effectively, in Turkish and English language. | |||||
| 8 | Develops ability to reach information on different subjects required by the wide spectrum of applications of mechatronics engineering, criticize, assess and improve the knowledge-base; gains consciousness on the necessity of improvement and sustainability as a result of life-long learning; gains ability for monitoring the developments on science and technology; develops awareness on entrepreneurship, innovative and sustainable development and ability for continuous renovation. | |||||
| 9 | Gains ability to be conscious on professional and ethical responsibility, competency on improving professional consciousness and contributing to the improvement of profession itself. | |||||
| 10 | Gains knowledge on the applications at business life such as project management, risk management and change management and competency on planning, managing and leadership activities on the development of capabilities of workers who are under his/her responsibility working around a project. | |||||
| 11 | Gains knowledge about the global, societal and individual effects of mechatronics engineering applications on the human health, environment and security and cultural values and problems of the era; develops consciousness on these issues and develops awareness of legal results of engineering solutions. | |||||
| 12 | Gains the competence on defining, analyzing and surveying databases and other sources, proposing solutions based on research work and scientific results and communicate and publish numerical and conceptual solutions. | |||||
| 13 | Gains conciousness on the environmental and social responsibility and develops conciousness to be an individual in society. Gains ability to develop and implement projects and asses them with a critical view for their social implications and gains ability to change the related norms if necessary. | |||||
| 14 | Gains the competence on developing strategy, policy and application plans on the mechatronics engineering and evaluating the results in the context of quality standarts. | |||||
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 | ||