Scientific Computation syllabus

Algebra | Analysis | Applied & PDE | Prob & Stat | Scientific Computation | Topology

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• General Numerical Methods
• Numerical Linear Algebra
• Numerical solution of ordinary differential equations
• Finite difference methods for partial differential equations

General Numerical Methods

• Floating point arithmetic and round-off
  
• Solution of nonlinear equations.
  • Fixed point methods
  • Newton's method
    
• Interpolation and Polynomial approximation
  • Lagrange polynomials (and their Newton form)
  • Hermite interpolation
  • Cubic spline interpolation
    
• Numerical Differentiation and Integration
  • Trapezoid rule, Simpson's rule, etc., Gaussian quadrature
  • Richardson extrapolation

Numerical Linear Algebra

• Solutions of Ax=b
  • Direct methods - LU decompostion; Cholesky decomposition
  • Iterative methods - Jacobi, Gauss-Seidel, Conjugate Gradient
    
• Solution of least squares problem
  • Normal equations
  • QR factorization
    
• Eigenvalue problem
  • Power method
  • QR method for symmetric matrices

Numerical solution of ordinary differential equations

• Solution of initial value problem
  • Runge-Kutta methods
  • One-step methods
  • Multi-step methods
  • Consistency, stability and convergence
    
• Two-point boundary value problem.

Finite difference methods for partial differential equations

• Transport equation in one dimension
  • Solution of u _t = u _x
  • Consistency, stability and convergence
  • Von Neumann stability, amplification factor
  • CFL condition
    
• Heat equation
  • Solution in one dimension, u _t = u _xx , with various boundary conditions; consistency, stability and convergence; Von Neumann stability, amplification factor; Crank-Nicolson; implicit and explicit methods
  • Solution in rectangular domains in two and three dimensions. Crank-Nicolson; implicit and explicit methods, ADI
    
• Poisson's equation
  • Solution in rectangular domains in two and three dimensions. Direct and iterative methods.

References

[1] Numerical Analysis, 6th edition, by Richard L. Burden and J. Douglas Faires
[2] Finite difference schemes and partial differential equations by John C. Strikwerda
[3] Numerical linear algebra by Lloyd N. Trefethen and David Bau.
[4] Matrix computations by Gene H. Golub and Charles F. Van Loan.

Mathematics Department Tulane University 6823 St. Charles Ave New Orleans, LA 70118 phone: (504) 865-5727 fax: (504) 865-5063

Last Updated: July 19, 2005