Linear Algebra and Interpolation
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Jan 11 - Jan 15
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Banded matrices and Gauss elimination, operation counts, LU decomposition, round-off error, ill-conditioned matrices, stiffness, Cayley-Hamilton theorem
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Jan 18 - Jan 22
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Lagrange polynomials, polynomial interpolation, splines
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Numerical Differentiation
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Jan 25 - Jan 29
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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
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Trapezoidal rule, Simpsons rule, error analysis and mid-point rule
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Feb 08 - Feb 12
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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
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Initial value problems, numerical stability analysis, model equation, Runge-Kutta formulas, multi-step methods
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Feb 22 - Feb 26
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Implicit methods, phase and amplitude errors, system of differential equations, stiffness
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Feb 29
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Linearization of implicit solution of non-linear differential equations, boundary value problems, shooting method, direct methods
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Mar 02
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Midterm exam
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Mar 07 - Mar 11
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Spring break
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Mar 14 - Mar 18
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Non-uniform grids, eigenvalue problems
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Numerical Solution of Partial Differential Equations
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Mar 21 - Mar 25
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Classification of partial differential equations (PDEs), semi-discretization, Modified wavenumber analysis
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Mar 28 - April 01
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von Neumann stability analysis, modified equation analysis, Alternate direction implicit (ADI) methods and approximate factorization
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April 04 - April 08
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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
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Discrete Fourier series, fast Fourier transform (FFT), discrete sine and cosine transforms, Fourier spectral method, discrete Chebyshev transform
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April 18
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Introduction to unstructured grids and finite volume (FV) methods
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April 20
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Review for Final Exam
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April 25 - April 29
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Project Presentations and submission of reports
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TBD
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Final Exam
Time: TBD, Location: TBD
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