[Warlord]

Hello folks,

Does anyone know how to derive the formulas implemented for the EulerD2dt2Scheme::fvmD2dt2? Specially for the first one, applied when the mesh is moving. Any reference?

template<class Type>
tmp<fvMatrix<Type> >
EulerD2dt2Scheme<Type>::fvmD2dt2
(
    const dimensionedScalar& rho,
    GeometricField<Type, fvPatchField, volMesh>& vf
)
{
    tmp<fvMatrix<Type> > tfvm
    (   
        new fvMatrix<Type>
        (   
            vf, 
            rho.dimensions()*vf.dimensions()*dimVol
            /dimTime/dimTime
        )   
    );  

    fvMatrix<Type>& fvm = tfvm();

    scalar deltaT = mesh().time().deltaT().value();
    scalar deltaT0 = mesh().time().deltaT0().value();

    scalar coefft   = (deltaT + deltaT0)/(2*deltaT);
    scalar coefft00 = (deltaT + deltaT0)/(2*deltaT0);

    scalar rDeltaT2 = 4.0/sqr(deltaT + deltaT0);

    if (mesh().moving())
    {   
        scalar halfRdeltaT2 = 0.5*rDeltaT2;

        scalarField VV0 = mesh().V() + mesh().V0();

        scalarField V0V00 = mesh().V0() + mesh().V00();

        fvm.diag() = rho.value()*(coefft*halfRdeltaT2)*VV0;

        fvm.source() = halfRdeltaT2*rho.value()*
        (
            (coefft*VV0 + coefft00*V0V00)
           *vf.oldTime().internalField()

          - (coefft00*V0V00)*vf.oldTime().oldTime().internalField()
        );
    }
    else
    {
        fvm.diag() = (coefft*rDeltaT2)*mesh().V()*rho.value();

        fvm.source() = rDeltaT2*mesh().V()*rho.value()*
        (
            (coefft + coefft00)*vf.oldTime().internalField()
          - coefft00*vf.oldTime().oldTime().internalField()
        );
    }

    return tfvm;
}

Thank you