  The $\lambda$ and $m_{\beta\beta}$ mechanisms of neutrinoless double beta
decay ($0\nu\beta\beta$) occur with light neutrino exchange via $W_L-W_R$, and
$W_L-W_L$ mediation, respectively. In the present study, we calculate the
nuclear matrix elements (NMEs) for the $m_{\beta\beta}$ and $\lambda$
mechanisms of $0\nu\beta\beta$, which has origin in the left-right symmetric
model with right-handed gauge boson at TeV scale. The NMEs are calculated for
one of the $0\nu\beta\beta$ decaying isotope $^{48}$Ca in the interacting
nuclear shell-model using the GXPF1A effective interaction of $pf$-shell. The
NMEs are calculated in both closure and nonclosure approaches using four
different methods: closure, running closure, running nonclosure, and mixed
methods. All the NMEs are calculated incorporating the effects of the finite
size of nucleons and the revisited higher order terms such as isoscalar and
weak magnetism terms of the nucleon currents. Inclusion of the short-range
nature of nucleon-nucleon interaction in Miller-Spencer, CD-Bonn, and AV18
parametrizations is also taken care of. The comparative dependence of the
running closure and running nonclosure NMEs with the spin-parity of the allowed
states of intermediate nucleus $^{48}$Sc, the coupled spin-parity of the two
initial decaying neutrons and the final two protons, the cutoff excitation
energy of $^{48}$Sc, the cutoff number of states of $^{48}$Sc are also
examined. Results show that there are about 2-20\% enhancements in different
types of total NMEs, calculated in the nonclosure approach as compared to the
closure approach. The significant enhancements are found in the $M_{qGT}$ and
$M_{qT}$ type NMEs for the inclusion of the higher-order terms of the nucleon
currents.
