AbstractA Chebyshev collocation method for solving the unsteady two‐dimensional Navier–Stokes equations in vorticity–streamfunction variables is presented and discussed. The discretization in time is obtained through a class of semi‐implicit finite difference schemes. Thus at each time cycle the problem reduces to a Stokes‐type problem which is solved by means of the influence matrix technique leading to the solution of Helmholtz‐type equations with Dirichlet boundary conditions. Theoretical results on the stability of the method are given. Then a matrix diagonalization procedure for solving the algebraic system resulting from the Chebyshev collocation approximation of the Helmholtz equation is developed and its accuracy is tested. Numerical results are given for the Stokes and the Navier–Stokes equations. Finally the method is applied to a double‐diffusive convection problem concerning the stability of a fluid stratified by salinity and he