Delaware State University
Department of Applied Mathematics and Theoretical Physics
Dover, DE 19901
Numerical Methods for Partial Differential Equations
60-854-00,
3 cr.
Text:
K. Morton & D. Mayers:
Numerical Solution of Partial Differential Equations
Recommended:
S. Brenner & L. Scott:
J. Thomas:
The Mathematical Theory of Finite Element Methods
Numerical Partial Differential Equations
Numerical methods for Partial Differential Equations (PDEs) are a part of the problem solving
skills that are expected to be mastered by most of the university graduates working in a quantitative
field. The same fundamental concepts of convection, diffusion, dispersion and non-linearity are used to
simulate applications in physics, economics, biology, engineering and social sciences. Quantitative
answers for the real world can generally be obtained only from computations. The goal of this course is
to provide a basic foundation in numerical methods for PDEs include finite difference method and finite
element method.
Prerequisite: Some introductory level courses of Numerical Analysis, Differential Equations and
familiarity with one programming language such as Matlab, C++, Fortran, etc.
Topical schedule:




Introduction
Finite Difference
1. Parabolic equations in one space variable
2. Fourier analysis of the error
3. Parabolic equations in two and three dimensions
4. Hyperbolic equations in one space variable
5. The CFL (Courant-Friedrichs-Lewy) condition
6. Consistency, convergence and stability
7. Linear second order elliptic equations
Finite Element
1. Sobolev Spaces
2. Variational Formulation of Elliptic Equations
3. The Ritz-Galerkin Method and Some Finite Elements
4. Approximation Properties
5. Error Bounds for Elliptic Problems
Iterative Methods
1. Basic iterative schemes in explicit form
2. Optimal SOR (successive overrelaxation method) parameters
3. The conjugate gradient method
4. Multigrid Method
CURRICULUM COURSE REVIEW:
Numerical PDEs
1. Course Title/Number:
Numerical Methods for Partial Differential Equations/60-854-00
2. Number of Credits:
3
3. Curriculum Program Title:
Ph.D. in Applied Mathematics and Theoretical Physics
4. Curriculum/Course is:
[X]
New
[
[
Required Course
[X]
]
]
Revised
Elective Course
5. List Prerequisites:
Numerical Analysis (25-661) and some programming languages such as Matlab, C++, etc.
6. List Courses Being Replaced or Changed:
This is a new course.
7. List Courses Being Deleted:
No courses are being deleted.
8. Needs Statement:
Numerical methods for Partial Differential Equations (PDEs) are needed for almost all areas of
applied mathematics. It is a part of the problem solving skills that are expected to be mastered by most
of the university graduates working in a quantitative field. The same fundamental concepts of
convection, diffusion, dispersion and non-linearity are used to simulate applications in physics,
economics, biology, engineering and social sciences. Quantitative answers for the real world can
generally be obtained only from computations.
9. Catalog Description of the Course:
Scientists often model physical systems with partial differential equations. Analytic solutions
exist only for the most elementary PDEs; the rest must be tackled with numerical methods including the
finite difference and the finite element methods. The finite difference method (FDM) approximates the
solution of a PDE at a number of points in space. The finite element method (FEM) attempts to find the
best approximate solution to a PDE among some finite set of simple functions. In the end, the iterative
methods are included to solve linear equations obtained from either FDM or FEM.
CURRICULUM COURSE REVIEW:
Numerical PDEs
10. List of Objectives of the Course:
(1) To provide the students an introduction to numerical approximations of PDEs. (2) To train
the students to write programs to solve PDEs numerically. (3) To develop analytical skills for numerical
schemes of different type of PDEs.
11. Course Outline:
See the “Topical schedule” section in the attached brief syllabus.
12. Show how the proposed course fits into the curriculum or course sequence:
This course is very important as we outlined above. The prerequisite, numerical analysis (25661), is an existing course in the curriculum of applied mathematics. It relates curriculum courses are
scientific computation (25-821) and Partial Differential Equations (25-853). Together, they will provide
the students a solid research background in applied mathematics.
13. Are there comparable courses in other departments?
No.
14. How will the students be affected by this course change?
This course provides the students an opportunity to develop necessary skills for research related
to the research program of the Department of Applied Mathematics and Theoretical Physics. The
students will grasp the skill for modeling real world applications quantitatively. This course will
significantly improve students’ professional competence, employability in technical fields and ability to
carry out research in applied mathematics. Neither this course nor its prerequisites increase the total
number of semester hours in this curriculum or the number of credit hours required for graduation.
15. What effect will this new course have on College resource?
None: this course will not require new or additional resources or staffing.
16. How will the course benefit the College?
This course will provide fundamental numerical methods to approximate PDEs, which are used
to model phenomena in physics, economics, biology, engineering and even social sciences. Not
restricted to applied mathematics, students from other departments such as chemistry, physics and
agriculture will find it a useful course in their own research areas as long as the applications they study
are modeled by PDEs and they need quantitative information of them.
CURRICULUM COURSE REVIEW:
Numerical PDEs
17. How will the change affect the program?
It will be one of the electives specific to the Ph.D. program (concentration in applied
mathematics) in this department. It will improve the quality of the Ph.D. curriculum in applied
mathematics.
18. Evaluation of Student Performance:
Homework Assignments
40 %
Projects
40 %
End-of-Term Presentation
20 %
Course Structure: Two (2) 75-minute lectures per week.
References
1. D. Braess: Finite Elements: Theory, fast solvers, and applications in solid mechancs.
(Cambridge University Press, 1997)
2. L.C. Evans: Partial Differential Equations (American Mathematical Society, 1998)
3. A. Taflove & S. Hagness: Computational Electrodynamics, 2nd Edition (Artech House, Boston,
London, 2000)
4. J. Stoer & R. Bulirsch: Introduction to Numerical Analysis, 2nd Edition (Springer-Verlag, 1992)
5. P. Monk: Finite Element Methods for Maxwell’s Equations (Oxford Science Publications, 2003)
Submitted to the Curriculum Committee Department of Applied Mathematics and Theoretical Physics
by: Jiguang Sun, on 27th of November, 2007