Mathematics Home / Scientific Computation Qualifying Exams

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

**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

**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 Initial Value Problems**

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

**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