Spectral Methods

Spectral Methods for Differential Equations and Applications in Mechanics

Content

This course provides the students with a class of spectral methods that can be
effectively implemented using the Fast Fourier Transform package. The theory and the
application of the spectral methods for solving boundary value problems are presented
with numerical illustrations. The boundary value problems of elliptic, parabolic and hy-
perbolic types will be studied numerically and the numerical stability is also investigated.
The main content of the course can be summarized in the following points:

  • Summary of the finite difference method and a new view on it as the derivatives of
    the local polynomial interpolants.
  • Procedure to derive the spectral numerical derivatives on the periodic and non-
    periodic grid and investigation of their accuracy.
  • Numerical solution of the one-dimensional elliptic, parabolic, hyperbolic PDEs with
    periodic, homogeneous and non-homogeneous boundary conditions. The examples
    are taken from the well-known equations in mechanics and physics.
  • Numerical stability analysis of time-stepping schemes.
  • Extension of the solution method to the high-dimensional differential equations.
Materials
  • Complete notes on blackboard
  • Exercise material will be handed out in the exercises
Audience

Students of Civil Engineering, Simulation Technology, COMMAS and other interested listeners

Prerequisites

3 of 3 assignments

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