Flux limiters
Flux limiters prevent the development of undesirable artifacts, such as unbounded values or spurious oscillations. Discover the different limiters for SOU, QUICK, CDS and MUSCL schemes.
The concept of flux limiters is similar to Total Variation Diminishing (TVD) schemes that are used in gas dynamics [14]. A limited higher order estimate for the value ϕip at an integration point whose upwind and downwind neighbors are nodes C and D, respectively, is expressed according to the following equation:
where:
- The subscript HO denotes the unlimited higher-order approximation.
- βC,ip is the limiter 0 < βC,ip < 1.
The limiter is a blending between the UDS scheme and the higher order scheme. The flow solver selects the limiter as the largest possible value within the solution, which prevents the development of undesirable artifacts, such as unbounded values or spurious oscillations.
SOU, QUICK, and CDS limiters
The limiters used for the SOU, QUICK, and CDS schemes are based on the convective boundedness criterion [15]. The integration point value of ϕ is limited as a requirement of the convective boundedness criterion by:
The limiter is determined using the following equation:
where ϕip is the closest value to ϕip, HO.
When the high-resolution advection scheme is selected, the unlimited second-order numerical advection correction approximation is employed, which is based on the cell-centered gradient, ∇ϕC, at the upwind node:
In this case, the limiter is determined according to the following equations [27]:
where:
- δ is the average mesh spacing.
- ϕext is an extremal value over the stencil of the node C. Stencil is the geometric arrangement of a nodal group that relate to the node C.
- The factor K is a blending of the limited and unlimited solutions, and determines the magnitude of oscillations in the solution.
The maximization in the first equation is carried out over all integration surfaces bounding the control volume about node C.
MUSCL limiters
The Monotonic Upstream-centered Scheme for Conservation Laws (MUSCL) scheme uses the flux limiter function when approximating the value of ϕip.
The flux limiter function, Ψ(r) is based on the ratio, r [63], of the upwind-side gradient to downwind-side gradient, which is defined as:
The available limiter functions for the MUSCL scheme are:
The flux limiter functions exhibit second-order TVD behavior. They are designed to traverse a specific region of the solution, known as the TVD region, to ensure the stability of the scheme. Second-order TVD limiters adhere to the following criteria: