Doubt regarding MUSCL scheme implementation and associated limiters
 

Dear all,

My objective is to simulate heating profiles over a blunt body in viscous supersonic/hypersonic ideal gas flow. Presently in the first order version of my code, I am using a Finite Volume based Approximate Riemann solver code. For better accuracy I would like to make the spatial order of accuracy of the code second order. In this regard, I tried implementing van Leer's MUSCL reconstruction procedure and used certain limiters to make the reconstruction monotone. I used Strong Stability Preserving Runge Kutta method of Gottlieb et al [1] to get second order time accuracy.
This is where I am facing problems.

Doubt regarding MUSCL interpolation formula:

Firstly, in literature, MUSCL scheme is expressed in two forms ( consider only left values specified at an interface i-1/2 between Ui-1 and Ui):

Form1: UL = Ui-1 + 1/4 * phi_L* [(1+k)*(Ui-Ui-1) + (1-k)*(Ui-1-Ui-2) ]

Form2: UL = Ui-1 + 1/4 * phi_L* [(1+k*phi_L)*(Ui-Ui-1) + (1-k*phi_L)*(Ui-1-Ui-2) ]



For k=0; these forms are equivalent. While for k=-1, these are clearly not. Most books discuss form 1. While most papers (ex 2563 of ref [2] or pg 1392 of ref [3] ) use form2 usually with k=1/3. Lohner in his book [4] also uses form 2 ( sec 10.4.2 ).


Are they equivalent in all cases or are these forms applicable only to specific situations?

Doubt regarding associated limiter:

Secondly, the doubt is regarding the form of limiters used in conjunction with these forms. Consider two famous limiters:


van Albada form 1: 2R/(R^2+1)

van Albada form 2: R^2 + R/ R^2 + 1


min-mod form 1: min(2/1+R, 2R/1+R)

min-mod form 2: min(1,R)


Berger et al [5] refers to form1 as slope limiter while form 2 as flux limiter.
My understanding is that flux limiters are used directly on flux reconstruction while it is the slope limiters which are employed on the reconstructed solution values. I am aware that flux limiters can be converted into slope limiters if they have the symmetric property.


If that is so, in my particular case, I should be using only form 1 of these limiters for reconstruction of my state variables at a given interface? Will it be wrong if I use flux limiters instead?

Also, for specific value of k (kappa parameter in MUSCL scheme) are there prescribed limiters or can I use any limiter for any k value?

Specifically, if I want to use MUSCL scheme with k=-1 and min-mod limiter and k=1/3 with van Albada limiter, which MUSCL form and which limiter form should I use?


I would really appreciate any help from any kind soul on this doubt. Thanks in advance.

Regards,

[1] S. Gottlieb, C W Shu, E Tadmor, Strong stability preserving higher order time discretization methods, SIAM review, vol 43, 2001.

[2] H. NIshikawa, K. Kitamura, A Very simple, carbuncle free, boundary layer resolving, Rotated hybrid Riemann solver, JCP, 2008.

[3] Y X Ren, A robust shock capturing schemes based on rotated Riemann solvers, Comput and Fluids, 2003.

[4] 'Applied computational fluid dynamics techniques', R. Lohner, 2nd edition, Wiley, 2008.

[5] M Berger, M J Aftosmis, S M Murman, Analysis of slope limiters on unstructured grids, 43rd AIAA Aerospace meetings, 2005.