Diffusion Scheme for Discontinuous Data:
In finite-volume methods, the flux needs to be computed at
the control-volume interface. For advection, the flux is computed given the
left and right states extrapolated to the interface from the left and right
cells. For diffusion, however, the focus is typically on the direct evaluation of the
solution gradient at the interface.
Here, we consider a diffusion scheme that is defined by the interface flux
evaluated by the left and right
states just like the advection scheme.
Consider the linear diffusion equation
which can be written as
where the flux is defined by
We now consider solving this equation on a uniform mesh with
cell-averaged solution values.
Figure above shows reconstructed piecewise linear data commonly
used to construct second-order advection schemes (MUSCL schemes of Van Leer).
The left and right states at an interface, for example at face j+1/2, are
where we assume that the gradients are evaluated by the central-difference formula,
There is a useful diffusion scheme which can be directly applied to such data:
This is a second-order diffusion scheme.
The first term in the numerical flux is called the consistent term (consistently
approximates the diffusive flux) and the second term is
called the damping term (provides damping for high-frequency errors).
The damping term is very important, just like the dissipation is important for
advection schemes, particularly on highly-skewed grids in
We find from truncation error (or Fourier) analysis the following special values of the damping coefficient, :
Note that the 2h-Laplacian scheme is a scheme with zero damping:
the highest frequency cannot be damped (odd-even decoupling). The
central-difference is nice since the stencil is compact (the 5-point stencil reduces
to 3-point stencil automatically). The 4th-order scheme is nicer because it gives very
accuarte solutions on irregular grids due to the strong damping (if not
4th-order accurate any more on irregular grids).
1. 2h-Laplacian Scheme (bad scheme):
2. Central-Difference Scheme:
3. 4th-Order Scheme (recommended):
Anyway, the diffusion scheme that acts directly on the discontinuous data
allows an easy extension of an advection code to
an advection-diffusion code. Given the left and right states, we
compute the advective and diffusive fluxes at once.
See [ AIAA2010-5093 ], Section 4.1.3, page 8, for more
details on the above scheme (or CF2011).
See also [ Riemann solver for diffusion ] which proposes
a similar diffusion scheme.
( The alpha-scheme above is used in a practical CFD code [ The 28th Computational Fluid Dynamics Symposium C08-1 ], and
has been found to be robust for unstructured grids [ CF2014 ]. )
by Hiroaki Nishikawa, February 25, 2011.