Diffusion Scheme for Discontinuous Data:
In finitevolume methods, the flux needs to be computed at
the controlvolume 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
cellaveraged solution values.
Figure above shows reconstructed piecewise linear data commonly
used to construct secondorder advection schemes (MUSCL schemes of Van Leer).
The left and right states at an interface, for example at face j+1/2, are
given by
where we assume that the gradients are evaluated by the centraldifference formula,
.
There is a useful diffusion scheme which can be directly applied to such data:
,
where
.
This is a secondorder 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 highfrequency errors).
The damping term is very important, just like the dissipation is important for
advection schemes, particularly on highlyskewed grids in
multidimensions.
We find from truncation error (or Fourier) analysis the following special values of the damping coefficient, :
1. 2hLaplacian Scheme (bad scheme):

2. CentralDifference Scheme:

3. 4thOrder Scheme (recommended):

Note that the 2hLaplacian scheme is a scheme with zero damping:
the highest frequency cannot be damped (oddeven decoupling). The
centraldifference is nice since the stencil is compact (the 5point stencil reduces
to 3point stencil automatically). The 4thorder scheme is nicer because it gives very
accuarte solutions on irregular grids due to the strong damping (if not
4thorder accurate any more on irregular grids).
Anyway, the diffusion scheme that acts directly on the discontinuous data
allows an easy extension of an advection code to
an advectiondiffusion code. Given the left and right states, we
compute the advective and diffusive fluxes at once.
See [ AIAA20105093 ], 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 alphascheme above has been found to be robust and accurate for unstructured grids [ CF2014 ], and has been used in practical CFD codes
[ 2014 , AIAA 20203029 ] including a hypersonicflow solver [ AIAA 20221848 ].
by Hiroaki Nishikawa, February 25, 2011 (updated February 13, 2022).

