Scientific Computation Qualifying Exams

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