Limiters
Limiters, generally speaking, limit the slope when doing high-order (e.g. accuracy order greater than 1, e.g. non-constant polynomial) interpolations from finite volume cell centroids to faces. This limiting is done to avoid creating oscillations in the solution field in regions of steep gradients or discontinuities. Slope limiters, or flux limiters, are generally employed to make the solution Total Variation Diminishing (TVD). Borrowing notation from here, the Total Variation when space and time have been discretized can be defined as
where is the discretized approximate solution, denotes the time index, and . A numerical method is TVD if
Limiting Process
Borrowing notation from (Greenshields et al., 2010), we will now discuss computation of limited quantities, represented by where represents one side of a face, and represents the other side. To be clear about notation: the equations that follow will have a lot of and . When computing the top quantity (e.g. for ) we select the top quantities throughout the equation, e.g. we select for and for . Similarly, when computing bottom quantities we select the bottom quantities throughout the equation. We will also have a series of "ors" in the text. In general left of "or" will be for top quantities and right of "or" will be for bottom quantities.
Interpolation of limited quantities proceeds as follows:
where denotes the or cell centroid value of the interpolated quantity and
where represents a flux limiter function and
(1)
where is the norm of the distance from the face to the cell centroid and is the norm of the distance from the face to the cell centroid. Note that this definition of differs slightly from that given in (Greenshields et al., 2010) in which the denominator is given by , the norm of the distance between the and cell centroids. The definition given in Eq. (1) guarantees that . Note that for a non-skewed mesh the definition in Eq. (1) and (Greenshields et al., 2010) are the same.
The flux limiter function takes different forms as shown in Table 1. is computed as follows
where corresponds to the or cell centroid gradient and .
The following limiters are available in MOOSE. We have noted the convergence orders of each (when considering that the solution is smooth), whether they are TVD, and what the functional form of the flux limiting function is.
Table 1: Limiter Overview
| Limiter class name | Convergence Order | TVD | |
|---|---|---|---|
VanLeer | 2 | Yes | |
Upwind | 1 | Yes | 0 |
CentralDifference | 2 | No | 1 |
MinMod | 2 | Yes | |
SOU | 2 | No | |
QUICK | 2 | No |
References
- Christopher J Greenshields, Henry G Weller, Luca Gasparini, and Jason M Reese.
Implementation of semi-discrete, non-staggered central schemes in a colocated, polyhedral, finite volume framework, for high-speed viscous flows.
International journal for numerical methods in fluids, 63(1):1–21, 2010.[BibTeX]