Alternative schemes for advection attempt to overcome problems with boundedness and accuracy of the linear and upwind schemes respectively. Many schemes apply a limiter between from upwind and from the linear scheme Eq. (3.4 ) according to
Limited schemes attempt to optimise at each face, based on the local , to maximise accuracy whilst maintaining boundedness.
Many schemes analyse the change in gradient of between the face and upwind cell in the direction connecting cell centres. They deﬁne a function of a ratio of consecutive gradients as:
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 does not increase in time, “overshoots” and oscillations associated with unboundedness will not occur.
To qualify as TVD, the limiter function 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 functions tend signiﬁcantly to downwind, i.e. towards the upper part of the shaded area near .
A further property of a limited scheme is symmetry. A scheme is symmetric when the condition is satisﬁed. When this occurs, the scheme applies the same limiter to the gradient of , irrespective of the sign of its gradient. As a consequence, a property , initialised with a symmetric proﬁle, e.g. a bell curve, will retain its symmetry under advection.