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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