Numerical Methods in Engineering

24-703 and 12-703, Spring 2016

Satbir Singh
Sample project from 2014 (Montgomery, Saliba, Li): Planetary orbital motion simulation using RK4 method


Lecture:
Day and Time: Mondays and Wednesdays, 2:30 - 4:20 PM
Location: WEH 5409


Instructor Office Hours:
Time: Wednesday, 1:00 - 2:00 PM
Location: Scaife Hall (SH) 319


TA Office Hours:
Time: Tuesdays 5:00 - 6:00 PM, Fridays 3:00 - 4:00 PM
Location: Tuesdays SH 205, Fridays SH 206


Course Description:
Many real-world engineering problems are governed by mathematical equations that cannot be solved using analytical methods. In such cases, one has to rely on numerical methods to predict a system's behavior. This course emphasizes numerical methods for solving engineering problems using a computer. Students will be introduced to many common techniques that are used to find numerical solutions to integral and differential equations governing variety of physical phenomena. Since the course is at the intersection of math and computer programming, students will be required to develop computer algorithms, employ them to find solutions, and explain mathematical and physical behavior of those solutions. Students can use either MATLAB, FORTRAN, or C/C++ as programming languages. Teams of students will work on projects in which they will apply techniques learned in the course to develop a numerical solver for simulations of real-world applications.

Prerequisites: Undergraduate level Numerical Methods or equivalent

Textbooks:
  • Fundamentals of Engineering Numerical Analysis, Parviz Moin
  • Numerical Recipes, 2nd edition
    • Grading:
  • Homeworks (50%)
  • Project (20%)
  • Midterm exam (15%)
  • Final exam (15%)
    • Tentative Syllabus Outline:
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    Linear Algebra and Interpolation
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    Jan 11 - Jan 15 Banded matrices and Gauss elimination, operation counts, LU decomposition, round-off error, ill-conditioned matrices, stiffness, Cayley-Hamilton theorem
    Jan 18 - Jan 22 Lagrange polynomials, polynomial interpolation, splines
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    Numerical Differentiation
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    Jan 25 - Jan 29 Construction of finite difference schemes, order of accuracy, modified wavenumber analysis, matrix representation of finite difference schemes
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    Numerical Integration
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    Feb 01 - Feb 05 Trapezoidal rule, Simpsons rule, error analysis and mid-point rule
    Feb 08 - Feb 12 Romberg integration, Richardsons extrapolation, adaptive quadrature, Gauss quadrature
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    Numerical Solution of Ordinary Differential Equations
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    Feb 15 - Feb 19 Initial value problems, numerical stability analysis, model equation, Runge-Kutta formulas, multi-step methods
    Feb 22 - Feb 26 Implicit methods, phase and amplitude errors, system of differential equations, stiffness
    Feb 29 Linearization of implicit solution of non-linear differential equations, boundary value problems, shooting method, direct methods
    Mar 02 Midterm exam
    Mar 07 - Mar 11 Spring break
    Mar 14 - Mar 18 Non-uniform grids, eigenvalue problems
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    Numerical Solution of Partial Differential Equations
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    Mar 21 - Mar 25 Classification of partial differential equations (PDEs), semi-discretization, Modified wavenumber analysis
    Mar 28 - April 01 von Neumann stability analysis, modified equation analysis, Alternate direction implicit (ADI) methods and approximate factorization
    April 04 - April 08 Iterative methods for elliptic PDEs: point Jacobi, Gauss Seidel, SOR, steepest descent, conjugate gradient methods
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    Discrete Transform and Finite Volume Methods
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    April 11 - April 15 Discrete Fourier series, fast Fourier transform (FFT), discrete sine and cosine transforms, Fourier spectral method, discrete Chebyshev transform
    April 18 Introduction to unstructured grids and finite volume (FV) methods
    April 20 Review for Final Exam
    April 25 - April 29 Project Presentations and submission of reports
    TBD Final Exam
    Time: TBD, Location: TBD