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  • 9 - Finite-Difference Methods for Initial-Value Problems

  • from Part II - Numerical Methods
  • Kevin W. Cassel, Illinois Institute of Technology
  • Book:
    Matrix, Numerical, and Optimization Methods in Science and Engineering
    Published online:
    18 February 2021
    Print publication:
    04 March 2021, pp 466-526

  • Summary

    The application of finite-difference methods to initial-value problems, with emphasis on parabolic equations, is considered using the one- and two-dimensional unsteady diffusion equations as model problems.Single-step methods are introducted for ordinary differential equations, and more general explicit and implicit methods are articulated for partial differential equations.Numerical stability analysis is covered using the matrix method, von Neumann method, and the modified wavenumber method.These numerical methods are also applied to nonlinear convection problems.A brief introduction to numerical methods for hyperbolic equations is provided, and parallel computing is discussed.