Abstract of
Natural Finite Volume
Discretizations of the Gradient, Divergence, Laplacian, Diffusion
Tensor and Advective Term on Grids with Quadrilateral Cells
by Nicolas Robidoux
A new method of construction of compact conservative finite volume
divergences and gradients which compose into high quality Laplacians
and diffusion tensors is presented. A consistent treatment of
boundary conditions and advective terms is introduced. Over a wide
class of problems, the methods perform better than any other.
Discretizations of the Laplacian and diffusion operator with full
diffusion tensor with jumps at the cell boundaries are derived for
rough (possibly unstructured) grids with quadrilateral cells. With
compatible Dirichlet, Neumann and/or Robin boundary conditions, they
yield fluxes and solutions which converge at second order. The
method's convergence rates are not affected by discontinuities of
the source term, even when the jumps are not restricted to cell
boundaries.
The main ideas of the construction are as follows: Differential
forms theory, via Stokes' theorem, provides ``error free'' building
blocks for discretizations of the gradient and the divergence: the
natural gradient (via the Potential Theorem) and the natural
divergence (via the Divergence Theorem). These building blocks call
for fitted grid structures, one for the gradient and one for the
divergence. To obtain a Laplacian, an operator which converts
between discrete vector fields over the two grids must be available.
This operator---a discrete analog of the Hodge star operator of
differential forms theory---is constructed by solving a local
interpolation problem.
Numerical tests on logically rectangular grids, including a model of
diffusion through a sand-shale formation and an advection-diffusion
model problem with very small diffusion term, are used to
demonstrate the effectiveness and flexibility of the method.
Keywords:
Conservative finite volume schemes, second order
boundary conditions, Hodge star operator, self-adjoint discrete
Laplacians, full diffusion tensors, advective terms, rough grids
with quadrilateral cells, multivariate interpolation.
Complaints to
nrobidou@netscape.net (Nicolas Robidoux)
December 19, 2000.