@TechReport{ it:2003-003,
author = {Magnus Sv{\"a}rd and Jan Nordstr{\"o}m},
title = {A Stable and Accurate Summation-by-Parts Finite Volume
Formulation of the Laplacian Operator},
institution = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2003},
number = {2003-003},
month = jan,
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. Of particular interest are general 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 we 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.
}
}