# 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)
Consent of the department
Course Language English N/A Natural & Applied Sciences Master's Degree Face To Face Lecture, Question and Answer, Problem Solving. 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. The students who succeeded in this course; Choose and apply suitable finite difference methods for numerical solutions of partial differential equations encountered in science and engineering Discuss finite difference methods with respect to stability, convergence and consistency with a reasonable degree of mathematical rigor Solve linear systems arising from finite difference solutions of partial differential equations. Write and implement computer programs for the numerical solutions of partial differential equations by finite difference method. 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. 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

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 30 100

### Course Category

Core Courses X

### 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.
2 Has the ability to obtain, to evaluate, to interpret and to apply information by doing scientific research.
3 Can apply gained knowledge and problem solving abilities in inter-disciplinary research.
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.
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.
6 Can develop strategies, implement plans and principles on the area of study and can evaluate obtained results within the framework.
7 Can develop and extend the knowledge in the area and to use them with scientific, social and ethical responsibility.
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.
9 Has proficiency in English language and has the ability to communicate with colleagues and to follow the innovations in mathematics and related fields.
10 Has software and hardware knowledge in the area of expertise, and has proficient information and communication technology knowledge.
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.

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