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Scientific Computation Qualifying Exams

Syllabus Topics

This exam will cover the following topics:


  • General Numerical Methods
  • Numerical Linear Algebra
  • Numerical Methods for Ordinary Differential Equations
  • Numerical Methods for Partial Differential Equations

General Numerical Methods


  • Principles of Numerical Mathematics

    • Well-posedness and condition number of a problem
    • Stability and convergence of numerical methods
    • Machine representation of numbers

  • Rootfinding for Nonlinear Equations

    • The bisection, the secant and Newton's methods
    • Fixed-point iterations
    • Solution of nonlinear systems of equations

  • Polynomial Interpolation

    • Lagrange polynomials (and their Newton form)
    • Hermite interpolation
    • Approximation by splines

  • Numerical Differentiation and Integration

    • Finite-difference approximations of derivatives
    • Midpoint, trapezoidal, Simpson, Newton-Cote quadratures
    • Richardson extrapolation

  • Orthogonal Polynomials in Approximation Theory

    • Approximation of functions by Fourier series
    • Gaussian integration and interpolation
    • Fourier trigonometric polynomials

Numerical Linear Algebra


  • Fundamentals

    • Orthogonal vectors and matrices
    • Vector and matrix norms
    • The singular value decomposition (SVD)
    • Conditioning and condition number

  • Least Squares Problem

    • Normal equations
    • QR factorization

  • Solutions of Linear Systems of Equations

    • Direct methods - LU factorization; Cholesky factorization
    • Iterative methods - Jacobi, Gauss-Seidel, SOR, Conjugate Gradient

  • Eigenvalue Problem

    • Power method
    • QR method for symmetric matrices

  • Numerical Methods for Boundary Value Problems

    • Boundary value problems for ODEs
    • Boundary value problems for elliptic PDEs

Numerical Methods for Ordinary Differential Equations


  • Numerical Methods for Initial Value Problems

    • One-step methods
    • Linear multistep methods
    • Runge-Kutta methods
    • Consistency, stability and convergence

Numerical Methods for Partial Differential Equations


  • Finite-Difference Methods

    • Accuracy and derivation of spatial discretizations
    • Explicit and implicit schemes for parabolic equations
    • Consistency, stability and convergence, Lax equivalence theorem
    • Von Neumann stability, amplification factor
    • CFL condition for hyperbolic equations
    • Upwind schemes for hyperbolic equations
    • Leapfrog, Lax-Friedrichs and Lax-Wendroff schemes
    • Crank-Nicolson scheme for the heat equation
    • Discrete approximation of boundary conditions

  • Finite Element Methods: Derivation and Basic Properties
  • Finite Volume Methods: Derivation and Basic Properties
  • Splitting Methods

    • Dimensional splitting, ADI methods
    • Operator splitting methods for convection-diffusion equations

References

1. Numerical Analysis, 6th edition, by Richard L. Burden and J. Douglas Faires
2. An Introduction to Numerical Analysis, 2nd edition, by Kendall E. Atkinson
3. Numerical Mathematics, by Alfio Quarteroni, Riccardo Sacco and Fausto Saleri
4. Numerical Linear Algebra, by Lloyd N. Trefethen and David Bau
5. Matrix Computations, by Gene H. Golub and Charles F. Van Loan
6. Finite Difference Schemes and Partial Differential Equations, by John C. Strikwerda
7. Finite Difference Methods for Ordinary and Partial Differential Equations, by Randall J. LeVeque
8. Numerical Methods for Evolutionary Differential Equations, by Uri M. Ascher

Mathematics Department, 424 Gibson Hall, New Orleans, LA 70118 504-865-5727 math@math.tulane.edu