3.11 Limited advection schemes

Alternative schemes for advection attempt to overcome problems with boundedness and accuracy of the linear and upwind schemes respectively. Many schemes apply a limiter eqn between eqn from upwind and eqn from the linear scheme Eq. (3.4 ) according to

f = (1 ) U + L: \relax \special {t4ht=
When eqn, the scheme reduces to upwind and it becomes linear interpolation when eqn. For a uniform mesh (eqn), eqn represents interpolation using the downwind cell value.

Limited schemes attempt to optimise eqn at each face, based on the local eqn, to maximise accuracy whilst maintaining boundedness.

PICT\relax \special {t4ht=

Many schemes analyse the change in gradient of eqn between the face and upwind cell in the direction eqn connecting cell centres. They define a function of a ratio eqn of consecutive gradients as:

 r = max 2 -d---r----- 1;0 for scalar : j djrn f \relax \special {t4ht=
There are numerous published schemes that define the limiter eqn as a function of the gradient ratio eqn. Those schemes that are most useful are described in the following sections.

Total variation diminishing schemes

Many useful schemes fall into a class known as Total Variation Diminishing (TVD).7 The TVD idea is that if the total variation of field eqn does not increase in time, “overshoots” and oscillations associated with unboundedness will not occur.

PICT\relax \special {t4ht=

To qualify as TVD, the limiter function eqn must fall within the shaded area in a Sweby diagram (above).8 The TVD concept is a 1D analysis. For 3D CFD on irregular polyhedral meshes, oscillations are more likely to occur with TVD schemes whose eqn functions tend significantly to downwind, i.e. towards the upper part of the shaded area near eqn.

A further property of a limited scheme is symmetry. A scheme is symmetric when the condition eqn is satisfied. When this occurs, the scheme applies the same limiter to the gradient of eqn, irrespective of the sign of its gradient. As a consequence, a property eqn, initialised with a symmetric profile, e.g. a bell curve, will retain its symmetry under advection.

7Ami Harten, High resolution schemes for hyperbolic conservation laws, 1983.
8Peter Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws, 1984.

Notes on CFD: General Principles - 3.11 Limited advection schemes