IJPAM: Volume 41, No. 9 (2007)

SCIENTIFIC VISUALIZATION FOR PDES OF 2ND ORDER
III: COMPARATIVE VISUALIZATION OF SOLUTIONS
FOR ELLIPTIC, PARABOLIC, HYPERBOLIC
AND MIXED-TYPE CASES

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 work continues the realization of the scientific visualization program outlined in [#!PDEpaperA!#]. Here we address the comparative visualization of the exact solutions of initial-value and boundary-value problems for parabolic, hyperbolic and elliptic PDEs of second order. For this purpose we design comparable initial and boundary conditions for the three types and use colour (mode 1) mapping (see Section 2 of [#!PDEpaperA!#] and the references therein) to compare the solutions as 2D colour (or grey-scale) images. The knowledge accumulated in this first part of the paper is then used in its second part for the comparative visualization of the solutions of two model mixed-type problems. One of the observable differences is the diverse behaviour of the shock waves obtained due to the propagation of singularities along the characteristic cone in the hyperbolic part of the domain of the boundary-value problem. For the visualization in the second part of the paper we use again 2D colour (mode 1) mapping, but also provide equivalent 3D modelling of the solutions as surfaces, to show the changes on the boundary between the regions of parabolicity and hyperbolicity. Based on the graphical results, we provide some comparative analysis of the properties of the solutions in all cases. All graphical images are obtained via the in-house software developed as part of the work on [#!GundersenPhD!#] and [#!ODEpaper!#], see also [#!PDEpaperA!#], [#!PDEpaperB!#] and [#!PDEpaperD!#].

Received: November 11, 2007

AMS Subject Classification: 35A21, 35B30, 35J25, 35K05, 35K20, 35L05, 35L15, 35L20, 35M10, 65D18, 68U05, 34C30, 35B05, 35B65, 35E15, 35J15, 35K10, 35L10, 35M05, 35R05, 65D15, 65D30, 65D32, 68U20, 76M27

Key Words and Phrases: comparative visualization, 3-dimensional visualization, colour map, partial differential equation, parabolic, elliptic, hyperbolic, mixed-type, initial value, boundary value, isocurve, propagation of singularities

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