Different types of Flux Limiters in TVD schemes

Different types of Flux Limiters for TVD schemesPresented by :Anirudha Sant NA10B039K.V.Sumanth NA10B011Department of Ocean Engineering, I.I.T Madras.

Contents

TVD ConceptCriteria for TVD concept-r diagramsFlux limiter functionsComparison and properties

Finite Control Volume Method for Convection Diffusion EquationConsider the following convection diffusion equation :div(u) = div( grad ) + S being a general property Integration of control volume gives:

The previous equation gives flux balance problem : Convective flux and Diffusive flux

The principal problem in the discretization of the convective terms - the calculation of the value of transported property at control volume faces and its convective flux across these boundaries.Schemes of third-order and above have been developed for the discretization of convective terms with varying degrees of success.Problem with higher order schemes : Implementation of boundary conditions.The need of TVD scheme.TVD conceptGeneralization of higher-order scheme for transported quantity at the east face of a one-dimensional control volume.e = P + (E P)/2where is an appropriate function.We get an additional convective flux Fe(E P)/2

ContAfter inspecting various forms i.e. UD,CD,QUICK we say that value of function and the nature of the scheme is decided by the ratio of upwind-side gradient to downwind-side gradient (P W)/(E P)

Let with

ContGeneralized form for east face value e :

For the UD scheme (r) = 0For the CD scheme (r) = 1For the LUD scheme (r) = rFor the QUICK scheme (r) = (3 + r)/4

The function forvarious discretization schemes

Monotonicity PreserveringA desirable property for stable, non-oscillatory, higher-order schemes.What do we mean by preserving monotonicity?it must not create local extremathe value of an existing local minimum must be non-decreasing and that of a local maximum must be non-increasing.

Total VariationIts an implication of monotonicity-preserving criterion.How is it defined? For discrete data:

TV() = |2 1| + |3 2| + |4 3| + |5 4 |=|3 1| + |5 3 |Monotonicity preserving criterion:TV( n +1) TV( n )

Criteria for TVD schemeNecessary and sufficient conditions for a scheme to be TVD in terms of the r relationship: If 0 < r < 1 the upper limit is (r) = 2r, so for TVD schemes (r) 2r If r 1 the upper limit is (r) = 2, so for TVD schemes (r) 2Figure shows the shaded TVD region in a r diagram

Limiter FunctionThe idea of designing a TVD schemeLimit the range of possible values of the additional convective flux Fe(r)(E P)/2function (r) is called a flux limiter function.Choice of (r) tell the order of the scheme and its boundedness properties.Different flux limiters propertiesAs introduced by Sweby:The flux limiter function of a second-order accurate scheme should pass through the point (1, 1) in the r diagramIf 0 < r < 1 the lower limit is (r) = r, the upper limit is (r) = 1, so for TVD schemes r (r) 1 If r 1 the lower limit is (r) = 1, the upper limit is (r) = r, so for TVD schemes 1 (r) rChoice of (r)!!!

Region for second order TVD scheme

Symmetric properties for flux limitersSweby introduced symmetric properties for flux limiters:

What does it indicate?

Flux limiter functions

Flux limiters

All limiterfunctions in a r diagram

A class of flux limiters which include both extremes of the upper and lower boundaries may be defined as

which as , varies from 2 to 1 moves across the whole region from top to bottom and when

Swebys expressionIt is relatively easy to verify that Leonards QUICK limiter function is the only one that is non-symmetric, whereas all the others are symmetric limiters.Lien and Leschziners UMIST limiter function was designed as a symmetrical version of the QUICK limiter.Swebys expression is a generalisation of the Min-Mod and SUPERBEE limiters by means of a single parameter . The limiter becomes the Min-Mod limiter when = 1 and the SUPERBEE limiter of Roe when = 2. To stay within the TVD region we only consider the range of values 1 2.

All the limiter functions stay inside the TVD region and pass through the point (1, 1) on the r diagram, so they all represent second-order accurate TVD discretisation schemes.

Van Leer and Van Albadas limiters are smooth functions, whereas all the others are piecewise linear expressions.The Min-Mod limiter function exactly traces the lower limit of the TVD region, whereas Roes SUPERBEE scheme follows the upper limit.Evaluation of TVD schemesTVD schemes are generalisations of existing discretisation schemes, so they inherently satisfy all the necessary requirements of transportiveness, conservativeness and boundedness.Problem: 2D source-free pure convection of a transported quantity with the flow at 45 to the lines of a 50 50 grid.

The exact solution to the above problem is a step function at x 0.7.Comparison of two TVD schemes: Van Leer and Van Albada with UD and QUICK

It can be seen that TVD solutions show far less false diffusion than the UD scheme and are almost as close to the exact solution as the QUICK scheme.They do not show any non-physical overshoots and undershoots. The two TVD solutions are quite close to each other. The more complex limiter functions take up more computer CPU time.

Compared with an ordinary scheme, a calculation employing any TVD scheme would require more CPU time due to additional calculation overhead associated with evaluating the extra source termsFor example, the UMIST scheme, was found to require 15% more CPU than the standard QUICK scheme.The main advantage is that a TVD scheme guarantees wiggle-free solutions and accurate solution.There is no convincing argument in favour of any particular TVD scheme and the choice appears to be a matter of individual preference.

References

1D_convection_diffusion_equation_from_CFD_book_by_Versteeg_Malalasekara.

High Resolution Schemes Using Flux Limiters for Hyperbolic Conservation Laws. P. K. SwebySIAM Journal on Numerical Analysis, Vol. 21, No. 5. (Oct., 1984), pp. 995-1011.StableURL: http://links.jstor.org/sici?sici=00361429%28198410%2921%3A5%3C995%3AHRSUFL%3E2.0.CO%3B2-Y.THANK YOU