Computational Science and Engineering Track
Purpose of the Program
The purpose of this program is to meet the needs of students who want to learn the basic concepts and skills of Computational Science and Engineering and then continue toward a Ph.D. in a discipline outside Applied Mathematics. A student who completes this program successfully will obtain a master’s degree in Applied Mathematics, in the Computational Science and Engineering Track.
The program is designed to provide interested students with a foundation in computational mathematics and, at the same time, to allow sufficient latitude for students to become proficient in an outside discipline. Approximately half of the credits for the master’s degree will be taken from a department other than Applied Mathematics.
Admission to the Program
A student in the Computational Science and Engineering Track will be enrolled simultaneously in two graduate programs, one in Applied Mathematics and one in the department from which the student wishes to receive a Ph.D. An interested student may apply for admission to this track either when applying for graduate study at CU or at any time during the student’s first two years of graduate study. First-year and second-year graduate students in any of the participating departments may apply for admission to this program.
Curriculum
The proposed curriculum is flexible in that a student may choose from a set of courses most useful to the discipline in which the Ph.D. is sought. Each participating department will have a specific set of appropriate courses that will be specified in a Memorandum of Understanding between that department and the Department of Applied Mathematics.
Four examples of such course listings are given below (Refer to Some Sample Programs section) for: (A) Physics; (B) Astrophysical and Planetary Sciences (APS); (C) Aerospace Engineering; and (D) Atmospheric and Oceanic Sciences (PAOS).
Proposed Curriculum for Computational Science and Engineering Track
I. APPM courses - 15 credits:
1. APPM 5600 (3 credits) – Numerical Analysis 1
2. APPM 5610 (3 credits) – Numerical Analysis 2
3. APPM 5440 (3 credits) – Applied Analysis 1
(Students who have not had the equivalent of APPM 4440 and APPM 4450 may need to take these or comparable courses before taking APPM 5440.)
4. One of the following choices, to be approved by the outside advisor:
APPM 5470 (3 credits) – Methods of Applied Mathematics: Partial Differential and Integral Equations
APPM 5460 (3 credits) – Methods in Applied Mathematics: Dynamical Systems, Differential Equations, and Chaos
APPM 5560 (3 credits) – Markov Processes, Queues and Monte Carlo Simulations
Other approved applied mathematics courses
5. Thesis Option:
APPM 6950 (3 credits) – Master’s Thesis
6. Non-thesis option
Graduate electives (6 credits), at least 3 credits in applied mathematics (A student choosing this option must pass one of the four APPM preliminary exams.)
II. CSCI courses – minimum of 3 credits:
CSCI 5576 (4 credits) – High-Performance Scientific Computing 1 (or other approved CSCI course)
III. Credits in second department - 12 credits approved by outside advisor
Thesis option: must include 3 credits of MS thesis in outside department
IV. Students working as TAs in APPM also must take:
APPM 7400 (1 credit) – Teaching Excellence Seminar
V. The Graduate School requires at least 30 graduate credits.
VI. To waive any of the course requirements:
a) A student can waive the requirement for APPM 5600 and APPM 5610 by passing the APPM prelim in Numerical Analysis.
b) A student can waive the requirement for APPM 5440 by passing the APPM prelim in Applied Analysis, or by passing the final exam in APPM 5440 given that year.
c) A student can waive the requirement for APPM 5470 by passing the APPM prelim in PDEs.
VII. Other recommended courses:
APPM 5430 (3 credits) – Methods in Applied Mathematics: Applications of Complex Variables
APPM 5480 (3 credits) – Methods of Applied Mathematics: Approximation Methods
APPM 6610 (3 credits) – Introduction to Numerical Partial Differential Equations
APPM 7400 (3 credits) – Topics in Applied Mathematics: Stochastic Simulations
Some Sample Programs
A. Computational Physics, non-thesis option
Any four of the following six courses, along with the required courses in APPM and CSCI:
PHYS 5250 (3 credits) – Introduction to Quantum Mechanics 1
PHYS 5260 (3 credits) – Introduction to Quantum Mechanics 2
PHYS 7310 (3 credits) – Electromagnetic Theory 1
PHYS 7320 (3 credits) – Electromagnetic Theory 2
PHYS 5210 (3 credits) – Theoretical Mechanics
PHYS 7230 (3 credits) – Statistical Mechanics
B. Computational Astrophysics, non-thesis option
Any four of the following eight courses, along with the required courses in APPM and CSCI:
ASTR 5110 (4 credits) – Atomic and Molecular Processes
ASTR 5120 (4 credits) – Radiative and Dynamical Processes
ASTR 5140 (3 credits) – Astrophysical and Space Plasmas
ASTR 5150 (3 credits) – Introductory Plasma Physics
ASTR 5400 (3 credits) – Introduction to Fluid Dynamics
ASTR 5540 (3 credits) – Mathematical Methods
ASTR 5560 (3 credits) – Radiative Processes in Planetary Atmospheres
ASTR 5820 (3 credits) – Origin and Evolution of Planetary Systems
C. Computational Aerospace Mechanics, non-thesis option
Fall semester, first year
ASEN 5012 (3 credits) – Mechanics of Aerospace Structures
APPM 5470 (3 credits) – Methods of Applied Mathematics: Partial Differential and Integral Equations
APPM 5600 (3 credits) – Numerical Analysis 1
Spring semester, first year
ASEN 5022 (3 credits) – Dynamics of Aerospace Structures
APPM 5610 (3 credits) – Numerical Analysis 2
CSCI 5576 (4 credits) – High-Performance Scientific Computing
Fall semester, second year
ASEN 5007 (3 credits) – Introduction to Finite Elements
APPM 5440 (3 credits) – Applied Analysis 1
APPM Elective (3 credits)
Spring semester, second year
ASEN Core (3 credits)
ASEN Elective (3 credits)
Elective in any department (3 credits)
Note: A student may replace APPM 5470 (PDEs) with either APPM 5460 (Dynamical Systems) or APPM 5560 (Markov Processes).
Any four of the following seven courses, along with the required courses in APPM and CSCI:
ATOC 5060 (3 credits) – Dynamics of the Atmosphere
ATOC 5061 (3 credits) – Dynamics of Oceans
ATOC 5400 (3 credits) – Introduction to Fluid Dynamics
ATOC 5410 (3 credits) – Fluid Instabilities, Waves, and Turbulence
ATOC 5560 (3 credits) – Radiative Processes in Planetary Atmospheres
ATOC 6100 (3 credits) – Predicting Weather and Climate
Master’s Requirements
In accordance with the rules of the Graduate School, credits from a given graduate course can be applied toward only one master’s degree.
Specifically, if graduate credits from a participating department are used to obtain a master’s degree in Applied Mathematics in the Computational Science and Engineering track, those same credits cannot also be used to obtain a master’s degree from the other department. The credits may be applied toward a Ph.D. from either department.
The program is self-contained. If desired, a student can leave the program with a master’s degree in Applied Mathematics and embark on a career at the master’s level.
