Advection-dispersion equation is solved in numerically by using combinations of differential quadrature method (DQM) and various time integration techniques covering some explicit or implicit single and multi step methods. Two different initial boundary value problems modeling conservative and nonconservative transports of some substance represented by initial data are chosen as test problems. In the first case, pure advection conservative model problem is studied. The second problem models motion of nonconservative substance and simulates fade out of it as time proceeds. The errors between analytical and numerical results are measured by discrete maximum norm. Comparison with some earlier works indicates that the proposed algorithms generate more accurate and valid results for some discretization parameters.