IJPAM: Volume 41, No. 9 (2007)

SCIENTIFIC VISUALIZATION FOR PDES OF 2ND ORDER IV:
COMPARISON BETWEEN THE CONTINUAL AND
THE DISCRETE BOUNDARY VALUE PROBLEM

Lubomir Dechevsky$^1$, Joakim Gundersen$^2$
$^{1,2}$R&D Group for Mathematical Modelling
Numerical Simulation and Computer Visualization
Institute for Information, Energy and Space Technology
Narvik University College
2, Lodve Lange's St., P.O. Box 385, Narvik, N-8505, NORWAY
$^1$e-mail: ltd@hin.no
$^1$url: https://ansatte.hin.no/ltd/
$^2$e-mail: joag@hin.no
$^2$url: https://ansatte.hin.no/joag/


Abstract.This paper completes the sequence [#!PDEpaperA!#], [#!PDEpaperB!#], [#!PDEpaperC!#], dedicated to the scientific visualization of Green's functions and solutions of initial-value and boundary-value problems for 2nd order linear PDEs. Here we compare the continual and the discrete (via finite-difference scheme) initial/boundary problem for the heat equation in two ways:

  • We study the shape of isocurves and the topology of isocurves and isosurfaces of the discrete Green's function and how they depend on the parameter $\sigma$ which controls the stability and is a measure of the implicitness of the difference scheme. The graphical results obtained can be compared to the graphical results in [#!PDEpaperB!#] for the continual Green's function of the exact continual initial/boundary problem for the heat equation.
  • We study the structure (size and smoothness) of the error of approximation via the difference scheme as a function of the parameter $\sigma$. Both stable and unstable cases are considered.

On the basis of the results obtained some comments and recommendations are made in the concluding section.

Received: November 11, 2007

AMS Subject Classification: 35K05, 65M10, 65N10, 68U05, 35C15, 35E15, 35G10, 35G15, 35K10, 35K15, 35K20, 65M05, 65M06, 65M15, 65N05, 65N06, 65N15, 65N20, 68U07, 68U20

Key Words and Phrases: scientific visualization, heat equation, initial value, boundary value, finite difference scheme, explicit, implicit, Crank-Nicholson difference scheme, backward Euler difference scheme, stability, convergence, Green's function, continual, discrete, isocurve, colour map, oversmoothing, singularity, computational geometry, computer graphics, applied mathematics

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2007
Volume: 41
Issue: 9