Fast contouring of solutions to partial differential equations

ABSTRACT

The application of Differential Equation Interpolants (DEIs) to the visualization of the solutions to Partial Differential Equations (PDEs) is investigated. In particular, we describe how a DEI can be used to generate a fine mesh approximation from a coarse mesh approximation; this fine mesh approximation can then be used by a standard contouring function to render an accurate contour plot of the surface. However, the standard approach has a time complexity equivalent to that of rendering a surface plot, O(fm²) for each element of the coarse mesh, (where fm is the ratio of the width of the coarse mesh to the fine mesh). To address this concern three fast contouring algorithms are proposed that compute accurate contour lines directly from the DEI, and have time complexity at most O(fm) for each coarse mesh element.

REFERENCES

• 1: Bradbury, E. L. 2002. Fast contouring of solutions to partial differential equations. M.S. thesis, Department of Computer Science, University of Toronto.
• 2: Burden, R. L. and Faires, J. D. 2001. Numerical Analysis, 7 ed. Brooks/Cole.
• 3: W. H. Enright, Accurate approximate solution of partial differential equations at off-mesh points, ACM Transactions on Mathematical Software (TOMS), v.26 n.2, p.274-292, June 2000 [doi>10.1145/353474.353482]
• 4: Thomas A. Grandine, Applications of Contouring, SIAM Review, v.42 n.2, p.297-316, June 2000 [doi>10.1137/S003614459936403X]
• 5: MathWorks. MATLAB online documentation, 12 ed. MathWorks.
• 6: Ramos, G. A. and Enright, W. H. 2001. Interpolation of surfaces over scattered data. Proceedings of the IASTED International Conference VISUALIZATION, IMAGING, AND IMAGE PROCESSING, 219--224.
• 7: Schiesser, W. 1991. The Numerical Method of Lines Integration of Partial Differential Equations. Academic Press.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.2 Approximation

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.7 Ordinary Differential Equations
      G.1.8 Partial Differential Equations
  G.4 MATHEMATICAL SOFTWARE

  J. Computer Applications
  J.2 PHYSICAL SCIENCES AND ENGINEERING

General Terms:
Algorithms, Performance

Keywords:
PDEs, contour, interpolation, rendering, visualization