A two-dimensional hydrodynamic code has been developed and used for the study of thin accretion discs. It is found that, in the absence of strong viscous dissipation, tidally induced waves spiral in discs all the way to the inner boundary. While this result was predicted analytically, numerical studies have always had too much dissipation to validate the prediction. The growth of high order unstable modes has also been followed in the non-linear regime, until the flow undergoes a subcritical transition to turbulence. For the first time in the history of accretion discs, hydrodynamic turbulence has been obtained in a numerical study (i.e. except for convection and MHD turbulence).