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 | Elective Courses |
| Course Level | Ph.D. |
| 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 | X |
|---|---|
| Major Area Courses | |
| 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 | Alanında, bağımsız olarak, bir problem kurgulayabilir, çözüm yöntemi geliştirerek problemi çözebilir ve sonuçları değerlendirebilir | X | ||||
| 2 | Matematiğin temel alanlarında ve kendi uzmanlığı olarak seçtiği alanda gerekli alt yapıyı oluşturur. | X | ||||
| 3 | Matematik literatürünü ve özel olarak kendi araştırma konusu ile ilgili ulusal ve uluslararası güncel yayınları takip edebilir ve bunlardan kendi araştırma konusu ile ilgili olanları çalışmalarında kullanabilir | X | ||||
| 4 | Bilimsel etik değerleri ve kuralları dikkate alır ve mesleki ve toplumsal yaşamda kullanabilir | X | ||||
| 5 | Kendi çalışmalarının sonuçlarını veya belli bir konudaki güncel çalışmaları ve bulguları, çeşitli bilimsel toplantılarda topluluk önünde Türkçe ve İngilizce olarak sunabilir ve tartışmalara katılabilir. | X | ||||
| 6 | Gerek bireysel, gerek bir çalışma grubunun üyesi olarak çalışabilme becerisini geliştirir | X | ||||
| 7 | Yaratıcı ve eleştirel düşünme, problem çözme, özgün bir çalışma üretme becerisini geliştirir. Bilimsel gelişmeleri takip eder, özümsediği bilgilerin analiz, sentez ve değerlendirmesini yapabilir. | X | ||||
| 8 | Kazandığı bilgi, beceri ve yetkinlikleri yaşam boyu geliştirmeye açık olur. | X | ||||
| 9 | Alanında özümsediği bilgiyi ve problem çözme yeteneğini disiplinler arası çalışmalarda uygulayabilir; karşılaşılan problemleri matematiksel modellerle ifade ederek, matematiksel bakış açısı ile farklı çözüm yöntemleri önerir. | X | ||||
| 10 | Matematik temelli yazılımları, bilişim ve iletişim teknolojilerini bilimsel amaçlı kullanabilir. | 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 | ||