  Grad's method of moments is employed to develop higher-order Grad's moment
equations---up to first 26-moments---for granular gases within the framework of
the (inelastic) Boltzmann equation. The homogeneous cooling state of a freely
cooling granular gas is investigated with the Grad's 26-moment equations in a
semi-linearized setting and it is shown that the granular temperature in the
homogeneous cooling state still decays according to Haff's law while the other
higher-order moments decay on a faster time scale. The constitutive relations
for stress and heat flux (the Navier--Stokes and Fourier relations) are
obtained by performing a Chapman--Enskog-like expansion on the Grad's 26-moment
equations and compared with those existing in the literature. The linear
stability of the homogeneous cooling state is analyzed through the Grad's
26-moment system and various sub-systems by decomposing them into longitudinal
and transverse systems. It is found that one eigenmode in both longitudinal and
transverse systems in case of inelastic gases is unstable. By comparing the
eigenmodes from various theories, it is established that the 13-moment
eigenmode theory predicts that the unstable eigenmode remains unstable for all
wavenumbers below a certain coefficient of restitution while any other
higher-order moment theory shows that this mode becomes stable above some
critical wavenumber for all values of coefficient of restitution. In
particular, the Grad's 26-moment theory leads to a smooth profile for the
critical wavenumber in contrast to the other considered theories. Furthermore,
the critical system size obtained through the Grad 26-moment and existing
theories are also in excellent agreement.
