course outline - The Ohio State University

MECHENG 6507: Intermediate Numerical Methods
Spring Semester, 2014
The Ohio State University
Scott Lab E0141
TuTh 3.55 pm — 5.15 pm
Instructor
Dr. Sandip Mazumder
Room E410, Scott Laboratory
Phone: 247-8099
Email: [email protected]
Office Hours
Room E410, Scott Laboratory, preferably by appointment, or walk-in.
Feel free to call instructor at home (614-442-5957) or on his cell phone (256-658-5004). No calls
after 10 pm please.
Textbook/ Reading Material
There is no required text for this course.
References for Supplementary Reading
 Numerical Mathematics and Computing, W. Cheney, and D. Kincaid, Second Edition, 1985,
Brooks Cole Publishing Company, ISBN 0534043569
 Numerical Solution of Partial Differential Equations: Finite Difference Methods, G.D. Smith,
Third Edition, 1985, Clarendon Press (Oxford), ISBN 0198596413
 Numerical Heat Transfer and Fluid Flow, S.V. Patankar, 1980, Hemisphere Publishing
Corporation, ISBN: 0891165223
Website
Go to ME/NE 6507 website on Carmen (carmen.osu.edu). Please check the website at least twice
a week for updates.
Lectures
Lecture notes (slides) will be posted on Carmen prior to the lecture. Students are encouraged to
print them out and bring to class. Lectures will also be recorded, and a link to the video will be
posted on Carmen approximately within 24 hours following the lecture.
Prerequisites
The instructor will assume that you are familiar with course material covered in undergraduate
numerical methods (ME 250 or equivalent). You will be expected to write computer programs
during the course and therefore, must be familiar with at least one computer language, such as
Fortran, Pascal, C/C++, or Matlab. If you are not familiar with any programming language, you
are advised to drop out of the course. If you use Matlab, you are allowed to use it as a
programming environment only. Use of Matlab in-built functions, unless otherwise stated, is
strictly prohibited in this course.
Grading Policy
Homeworks: 60%
Midterm Exam (in class): 20%
Final Exam (in class): 20%
Tentative Grading Scale
90% and above: A
80% to 90%: A70% to 80%: B+
60% to 70%: B
50% to 60%: BBelow 50%: C
Note: The above grading scale is meant to serve as a guideline, and may be shifted up or down.
Academic Misconduct
Please read http://studentaffairs.osu.edu/resource_csc.asp to understand your responsibilities as a
student. Failure to abide by the rules stated in this section is severely punishable by the
University.
Homework Policy
All homeworks will involve computer programming. You must submit a printout of your program
with the homework write-up. Even if your program is not working, you are encouraged to submit
it to earn partial credit. No late homework will be accepted, unless you are seriously ill. Any
attempt at using someone else’s program will be severely penalized, including expulsion from the
course. Electronic submission of homeworks is strongly discouraged. If you submit an
electronic copy, it must be submitted as a single self-contained PDF or MSWord file, and must
not contain any plots in color. No other format will be accepted.
Absence from Examination
If you do not appear for an examination without advance notice, you will automatically receive a
zero grade. Makeup exams will be entertained only if the instructor is notified at least one week
in advance.
Examinations
Both examinations will be open book/notes and in class. The format of the examination and other
relevant details will be discussed in class prior to the examination.
Policy on Seeking Help
On many instances, you may require the instructor’s help to debug programs. To get help, you
must meet the instructor in person with either a hard copy of your program or the program on a
laptop/tablet (the latter is preferred). E-mails to the instructor stating that the program does not
work (with attached program) and a request to look at it will be ignored. The instructor will not
debug or run your program. That is the students’ responsibility.
Tentative Syllabus and Course Schedule
This course is intended to introduce students to the finite-difference and finite-volume methods for
solving canonical partial differential equations encountered in engineering. Students interested in the
finite-element method should take ME 5168.
Date
1/7 (Tu)
1/9 (Th)
1/14 (Tu)
1/16 (Th)
1/21 (Tu)
1/23 (Th)
1/28 (Tu)
1/30 (Th)
2/4 (Tu)
2/6 (Th)
2/11 (Tu)
2/13 (Th)
2/18 (Tu)
2/20 (Th)
2/25 (Tu)
2/27 (Th)
3/4 (Tu)
3/6 (Th)
3/18 (Tu)
3/20 (Th)
3/25 (Tu)
3/27 (Th)
4/1 (Tu)
4/3 (Th)
4/8 (Tu)
4/10 (Th)
4/15 (Tu)
4/17 (Th)
4/28 (M)
Topics Covered/Major Deadlines
Introduction, classification of PDEs
General discussion of methods for solving PDEs, types of meshes used etc., Derivation of
finite-difference equations in 1D
Errors in difference approximations, application of boundary conditions, matrix setup
Gaussian elimination, solution to tri-diagonal and other banded systems
Treatment of non-linear sources, residual calculation, solution of coupled non-linear
algebraic equations using Newton’s method
Finite-difference in 2D, sparse systems, introduction to iterative solvers, Jacobi method
Gauss-Seidel method, successive over-relaxation (SOR), line-by-line methods (ADI).
Incomplete LU decomposition, pre-conditioning: basic philosophy, Stone’s strongly
implicit method (SIP)
Method of steepest descent, conjugate gradient (CG) method
Correction form of equations, inertial damping, convergence analysis of various iterative
methods, Fourier decomposition of errors
Spectral radius of convergence and calculation procedure
Convergence analysis continue for various schemes
Multi-grid methods, basic philosophy, geometric multi-grid (GMG)
Multi-grid methods continued, algebraic multi-grid (AMG)
Summary of linear algebraic equation solvers
MIDTERM EXAMINATION
Errors in difference approximations re-visited, higher-order methods
Parabolic problems, time marching, forward and backward Euler methods, CrankNicholson method, stability vs. accuracy
Parabolic problems continued…, higher order explicit methods
Cylindrical coordinate system
Finite-volume method (FVM), basic philosophy, difference between FVM and FDM
FVM continued…
Finite-volume integration of elliptic PDEs on unstructured mesh, general formulation,
geometry calculations
Boundary conditions for unstructured FVM
Unstructured mesh continued…, code development details
Hyperbolic PDEs, key issues, fluid flow, introduction to Euler and Navier-Stokes
Equations
Miscellaneous topics (TBD)
Miscellaneous topics (TBD)
FINAL EXAMINATION, 6.00 to 7.45 pm