Stability of finite volume approximations for the Laplacian operator on quadrilateral and triangular grids

Magnus Svärd, Jan Nordström

Research output: Contribution to journalArticlepeer-review

36 Citations (Scopus)

Abstract

Our objective is to analyse a commonly used edge based finite volume approximation of the Laplacian and construct an accurate and stable way to implement boundary conditions for time dependent problems. Of particular interest are unstructured grids where the strength of the finite volume method is fully utilised. As a model problem we consider the heat equation. We analyse the Cauchy problem in one and several space dimensions and prove stability on unstructured grids. Next, the initial-boundary value problem is considered and a scheme is constructed in a summation-by-parts framework. The boundary conditions are imposed in a stable and accurate manner, using a penalty formulation. Numerical computations of the wave equation in two-dimensions are performed, verifying stability and order of accuracy for structured grids. However, the results are not satisfying for unstructured grids. Further investigation reveals that the approximation is not consistent for general unstructured grids. However, grids consisting of equilateral polygons recover the convergence.

Original languageEnglish
Pages (from-to)101-125
Number of pages25
JournalApplied Numerical Mathematics
Volume51
Issue number1
DOIs
Publication statusPublished - Oct 2004
Externally publishedYes

Keywords

  • Accuracy
  • Finite volume methods
  • Stability
  • Unstructured grids

ASJC Scopus subject areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

Fingerprint

Dive into the research topics of 'Stability of finite volume approximations for the Laplacian operator on quadrilateral and triangular grids'. Together they form a unique fingerprint.

Cite this