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MUSCL Interpolation and Limiters

    The previously discussed flux discretization schemes require that the solution is known at the cell face boundaries in order to evaluate the flux.  It is alternately possible to calculate the flux at the cell center and extrapolate it to the boundary, but limited research has indicated that this method is less desirable than calculating the flux at the interface boundaries by extrapolating the solution variables to the same cell faces.  So, with this need in mind, the solution available at the cell centers must make use of some kind of geometric interpolation to estimate the solution at the interface boundaries.
    There are a great many ways to do this, and the choice of interpolation scheme is an entire field of study in itself.  One must consider both the stenciling of the scheme (i.e. upwind, centered, downwind, a combination, etc.) and the resulting order of accuracy of the interpolation when considering a scheme.  The output of the code is highly dependent on this choice.  For Gryphon, one established choice has been selected that is known as Monotone Upwind Schemes for Scalar Conservation Laws (MUSCL).  MUSCL interpolation is actually a brand name for a whole type of reconstruction-evolution methods originally pioneered by Van Leer.  After his original work, many variations were invented, and all of these derivatives are still often referred to as "MUSCL."  Thus there are many different possibilities.  The reconstruction-evolution technique as programmed in Gryphon is given here.
    A one parameter family of upwind biased schemes can be defined for any given cell j as shown in eqns. (65) and (66):

   +1/2 cell face

+1/2 cell face

   -1/2 cell face

-1/2 cell face

This form of the scheme is a stencil consisting of three points, the cell center j, and the adjacent cell to each side, j+1 and j-1.  The parameter phi is not really a parameter.  It is a flag for either first order or higher order reconstruction.  First order reconstruction can be performed by setting phi equal to 0 -- regardless of the parameter value.  Setting phi equal to 1 results in higher order reconstruction, which is a useful feature of this method.  The valid range for kappa is between -1 and 1 inclusive.  A value of less than one results in an upwind biased scheme, while a value of exactly one results in a centered scheme.  Any value of kappa results in a second order reconstruction with the exception of 1/3. which results in a unique 3rd order accurate reconstruction.  Sample values of kappa employed by Gryphon are given in Table 2.

Table 2. One Parameter High Order MUSCL Reconstruction Schemes as Programmed in Gryphon

1st order
3rd order

    This form of the MUSCL reconstruction provides a straightforward, powerful, high order solution to the problem of interpolating the conservative independent variables.  There is only one exception.  In the presence of shocks, one finds that employing any of the schemes outlined using eqns. (65) and (66) and Table 2 other than first order results in large non-physical oscillations occuring near the shocks.  This is a numerical issue related to the sharp, sudden change in the independent variables.  Schemes that correctly model this effect are called Total Variation Diminishing, or TVD, schemes.  TVD is a mathematical property which means that as time progresses the sum of all the steps (differences) between adjacent points must remain the same or decrease.  Thus, the amount of "variation" and a TVD scheme starts with is the maximum that it can ever have.  Unfortunately, higher order finite volume schemes of this nature do not have the TVD property, and thus the non-physical oscillation are allowed to develop as iteration proceeds.
    The standard way around this is through the use of limiters.  Gradient limiters limit the forward and backward gradient terms of the above MUSCL reconstruction to force the scheme to have the TVD property in a region of strong shocks.  Limiters can perhaps more simply be seen as a tool to limit the ratio of the forward and backward gradients of a given cell to avoid a non-physical interpolation result.  One seeks to limit the slope of the forward and backward gradients in such a way that the interpolated points do not create a new minimum or maximum.  If, for example, at a given cell, the value of the conservative variable is increasing from left to right and from the j-1 index cell to the j+1 index cell, then one needs to enforce the conditions that the right cell face boundary must be less than or equal to the value in the j+1 cell to avoid a new maximum.  Likewise, the value at the left cell boundary must be greater than or equal to the j-1 cell value to avoid a new minimum.  These constraints are summarized in eqn. (67).



If one uses eqn. (65) for example to evaluate the variable at the left cell boundary, one finds that the following relation occurs subject to the constraint of eqn. (67), as given in eqn. (68) (by simply doing some algebra to simplify the inequality).

TVD limit

This inequality forms the basis for the MIN-MOD limiter.  Many other limiters are available, each with their own slightly unique approach to solving the problem of localized maxima and minima created by the interpolation stenciling.  Remember that this is a direct consequence of the TVD property.  In other words, a scheme that adhere's to the TVD property cannot -- by definition -- be capable of generating any new minima and maxima.  For the purposes of Gryphon, the MUSCL stenciling has been placed in a standard format given in eqn. (69) and (70).

left state limited form

right state limited form

In the stenciling formulas, the limited gradients are in all cases a function of the appropriate gradient ratios.  The gradient limiters can be selected from the list given in eqns. (71), (72), (73), and (74).  Note that all the limiters return 0 if the gradient ratio is less than zero (negative).  The case where the gradient switches direction indicates that there is a shock within the cell range and the local reconstruction is reduced for first order in this instance.

MIN-MOD Limiter

min mod limiter

Van Albada Limiter

Van Albada limiter

Van Leer Limiter

Van Leer limiter

Superbee Limiter

Superbee limiter

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