  The chiral magnetic wave (CMW) has been theorized to propagate in the
deconfined nuclear medium formed in high-energy heavy-ion collisions, and to
cause a difference in elliptic flow ($v_{2}$) between negatively and positively
charged hadrons. Experimental data consistent with the CMW have been reported
by the STAR Collaboration at the Relativistic Heavy Ion Collider (RHIC), based
on the charge asymmetry dependence of the pion $v_{2}$ from Au+Au collisions at
$\sqrt{s_{\rm NN}}$ = 27 to 200 GeV. In this comprehensive study, we present
the STAR measurements of elliptic flow and triangular flow of charged pions,
along with the $v_{2}$ of charged kaons and protons, as a function of charge
asymmetry in Au+Au collisions at $\sqrt{s_{\rm NN}}$ = 27, 39, 62.4 and 200
GeV. The slope parameters extracted from the linear dependence of the $v_2$
difference on charge asymmetry for different particle species are reported and
compared in different centrality intervals. In addition, the slopes of $v_{2}$
for charged pions in small systems, \textit{i.e.}, $p$+Au and $d$+Au at
$\sqrt{s_{\rm NN}}$ = 200 GeV, are also presented and compared with those in
large systems, \textit{i.e.}, Au+Au at $\sqrt{s_{\rm NN}}$ = 200 GeV and U+U at
193 GeV. Our results provide new insights for the possible existence of the
CMW, and further constrain the background contributions in heavy-ion collisions
at RHIC energies.
