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Numerical Integration and ODEs [M12]

The course introduces numerical methods for ordinary differential equations. At the end of the course students should be adept at choosing methods appropriate for a specific application, they should understand the problem of stiffness and its associated difficulties.

The contents are:

Numerical Integration

  • Mid-point rule, Trapezium rule and Simpsons rule
  • Newton-Cotes formulae
  • Gauss rule/ Gaussian quadrature
  • Peano kernel theorem
  • Composite rules and order of convergence
  • Lobatto / Radau rule
  • Higher dimensions and the dimensional effect
  • Monte Carlo Methods

ODEs

  • One-step methods : Euler, backward Euler, Trapezoidal rule, etc
  • Order and Convergence
  • Stiffness and A-stability
  • Adams methods
  • Backward differentiation formulae
  • Rational methods
  • Runge-Kutta methods
  • Milne device with predictor and corrector
  • Zadunaisky device for error estimation