On the Solution of Parabolic and Hyperbolic PDE's by the Method of Lines Approach

Abstract

This tutorial paper surveys the problems arising in the coding and utilization of general purpose packages for the solution of PDE (partial differential equation) problems. It is shown that such packages can never be as general as a package for ODE (ordinary differential equation) problems. There exist, however, enough solvable problems to justify the coding of such general purpose packages using robust methods, such as the method of lines. Some hints are given on how to select the following parameters:

  1. the optimal step size for the integration over time
  2. the optimal order of the integration algorithm
  3. the optimal class of integration algorithms
  4. the optimal grid width for the discretization in space
  5. the optimal order of the approximation formulae for the computation of the spatial derivatives.

In this formulation, the problem encompasses the ODE systems integration (parameters (1) to (3)). For this reason, the simpler ODE problem is being discussed first in a separate section of the paper.

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Last modified: January 23, 2006 -- © François Cellier