Detailed Course Outline

• Introduction (lectures 1-2)

  • History and milestone algorithms lecture 1

  • Basic concepts:
    • Numerical error
    • Convergence and order of accuracy
    • Stability and conditioning number

  • Basic techniques lecture 2
    • Asymptotic error expansion
    • Richardson extrapolation
    • Perturbation analysis
    • Some techniques to reduce round-off errors

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• Iterative methods for linear systems (lectures 3-4)
  • Introduction lecture 3

  • Classic iterative methods
    • Jacobi Method
    • Gauss-Seidel method
    • Successive Overrelaxation method (SOR)

  • Convergence analysis

  • Krylov subspace methods lecture 4
    • Steepest descent method
    • Conjugate gradient (CG) method
    • Preconditioning
    • GMRES for nonsymmetric matrix

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• Methods for nonlinear systems (lecture 5)
  • Introduction lecture 5

  • Newton's method

  • Quasi-Newton method

  • Convergence analysis

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• Numerical methods for ODEs (lectures 6-8)
  • Introduction lecture 6

  • Single-step method
    • Euler methods
    • Runge-Kutta methods

  • Stability and convergence analysis

  • Multi-step methods lecture 7

  • Extension to first order systems and high order equations lecture 8

  • Methods for stiff systems

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• Numerical methods for PDEs (lectures 9-12)
  • Introduction lecture 9

  • Finite difference method

  • Fast direct Poisson solvers lecture 10

  • Convergence analysis

  • Methods for time-dependent problems lecture 11

  • Finite element method lecture 12

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• Monte Carlo Method and its Applications lecture 13

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