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In the present work, we examine the role of central (C), spin-orbit (SO) and tensor (T) components of two-nucleon interaction in the nuclear matrix elements (NMEs) of the two-neutrino double beta decay ($2νββ$) and the light neutrino-exchange mechanism of neutrinoless double beta decay ($0νββ$) of $^{48}$Ca in closure approximation and nonclosure approach. The NMEs are calculated in the nuclear shell-model framework using two-nucleon effective interaction GXPF1A used for $pf$ shell. The decomposition of the shell model two-nucleon interaction into its individual components is performed using the spin-tensor decomposition (STD). The NMEs for $2νββ$ are calculated in running nonclosure method. The NMEs for $0νββ$ are calculated with four different methods, namely, closure, running closure, running nonclosure, and mixed method. Results show that the magnitude of NMEs for $2νββ$ decreases about 7\% with the C+SO component of the interaction as compared to the C component. The magnitude of NMEs is further decreased about 9\% by adding T component to the C+SO component. The NMEs of $0νββ$ calculated in running nonclosure method are enhanced by about 8-10\%, 8-10\%, and 9-12\%, respectively, as compared to corresponding NMEs calculated in running closure method with C, C+SO components and total (C+SO+T) GXPF1A interaction for different SRC parametrization. For both $2νββ$ and $0νββ$, the NMEs calculated with C+SO component is in opposite phase with the NMEs calculated with C component and the total GXPF1A interaction.