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Numerical Differentiation and PDEs [M12]

The course introduces numerical methods for numerical Differentiation and their application to partial differential equations. At the end of the course students should be adept at choosing methods appropriate for a specific application and constructing their own methods, they should understand the various notions of stability.
The contents are:

Numerical Differentiation

  • Floating point arithmetic
  • Finite Differences
  • Forward, backward, centred divided differences
PDEs
  • Classification of PDEs
  • Parabolic PDEs
  • Boundary Problems and the Eigenvalue Analysis of Stability
  • Cauchy problems and the Fourier Analysis of Stability
  • Spectral methods
  • Elliptic PDEs
  • Computational Stencils and the associated algebraic systems
  • Hockney Algorithm
  • Multigrid methods
  • Splitting
  • Hyperbolic PDEs
  • Advection and wave equation
  • Finite elements