This paper presents a new meshless numerical scheme to overcome the problem of shock waves and to apply boundary conditions in cases of dam-break flows in channels with constant and variable widths. The numerical program solves shallow water equations based on the discrete mixed subdomain least squares (DMSLS) meshless method with collocation points. The DMSLS meshless method is based on the minimization of a least squares functional defined as the weighted summation of the squared residuals of the governing equations over the entire domain and requiring the summation of residual function to be zero at collocation points in boundary subdomains. The collocated discrete subdomain meshless method is applied on the boundary, whereas the collocated discrete least squares meshless technique is applied to the interior domain. The meshless scheme extends for dam-break formulation of shallow water equations. The model is verified by comparing computed results with analytical and experimental data for constant and varying width channels. The developed model is also used to study one-dimensional dam-break problems involving different flow situations by considering changes to the channel width, a bumpy channel with various downstream boundary conditions, and the effects of bed friction and bed slope as source terms on wave propagation. The accuracy of the results is acceptable.