SMART (bounded QUICK) scheme for OpenFOAM

SMART (Sharp and Monotonic Algorithm for Realistic Transport) algorithm is the bounded version of QUICK (Quadratic Upstream Interpolation for Convective Kinematics) algorithm proposed by Gaskell and Lau (1988)[1].

In this article, I used the modified version of SMART algorithm[2].

Modified SMART algorithm compared with QUICK algorithm in NVD diagram:

Ψ-r relationship of modified SMART algorithm is as follows:

Modified SMART algorithm compared with SuperBee, Minmod algorithms in Sweby's diagram:

I developed this modified SMART scheme based on SuperBee scheme in foam-extend-3.2.

Source codes:

SMART.H
#ifndef SMART_H
#define SMART_H

#include "vector.H"

// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //

namespace Foam
{

/*---------------------------------------------------------------------------*\
                       Class SMARTLimiter Declaration
\*---------------------------------------------------------------------------*/

template<class LimiterFunc>
class SMARTLimiter
:
    public LimiterFunc
{

public:

    SMARTLimiter(Istream&)
    {}

    scalar limiter
    (
        const scalar cdWeight,
        const scalar faceFlux,
        const typename LimiterFunc::phiType& phiP,
        const typename LimiterFunc::phiType& phiN,
        const typename LimiterFunc::gradPhiType& gradcP,
        const typename LimiterFunc::gradPhiType& gradcN,
        const vector& d
    ) const
    {
        scalar r = LimiterFunc::r
        (
            faceFlux, phiP, phiN, gradcP, gradcN, d
        );

        return max(min(min(4.0*r, (3.0 + r)/4.0), 4.0/3.0), 0);
    }
};


// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //

} // End namespace Foam

// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //

#endif

// ************************************************************************* //

SMART.C

#include "LimitedScheme.H"

#include "Limited01.H"

#include "SMART.H"

#include "DeferredCorrectionLimitedScheme.H"

// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //

namespace Foam

{

    makeLimitedSurfaceInterpolationScheme(SMART, SMARTLimiter)

    makeLimitedVSurfaceInterpolationScheme(SMARTV, SMARTLimiter)

    makeLLimitedSurfaceInterpolationTypeScheme

    (

        SMART01,

        Limited01Limiter,

        SMARTLimiter,

        NVDTVD,

        magSqr,

        scalar

    )

    // Deferred correction schemes

    makeDeferredSurfaceInterpolationScheme(SMARTDC, SMARTLimiter)

    makeDeferredVSurfaceInterpolationScheme(SMARTVDC, SMARTLimiter)

    makeLDeferredSurfaceInterpolationTypeScheme

    (

        SMART01DC,

        Limited01Limiter,

        SMARTLimiter,

        NVDTVD,

        magSqr,

        scalar

    )

}

// ************************************************************************* //

References:
1. P.H. Gaskell, A.K.C. Lau, Curvature-compensated convective transport; SMART, A new boundedness-preserving transport algorithm, International Journal for Numerical Methods in Fluids, Vol. 8, Iss. 6, pp. 617-641, 1988.
2. F. Moukalled, L. Mangani, M. Darwish, The Finite Volume Method in Computational Fluid Dynamics, Springer, pp. 440-443, 2015.