Context. We investigate the wave heating problem of a solar quiet region and present its plausible solution without involving shock formation. Aims. We aim to use numerical simulations to study wave propagation and dissipation in the partially ionized solar atmosphere, whose model includes both neutrals and ions. Methods. We used a 2.5D two-fluid model of the solar atmosphere to study the wave generation and propagation. The source of these waves is the solar convection located beneath the photosphere. Results. The energy carried by the waves is dissipated through ion-neutral collisions, which replace shocks used in some previous studies as the main source of local heating in quiet regions. Conclusions. We show that the resulting wave dissipation is sufficient to balance radiative and thermal energy losses, and to sustain a quasi-stationary atmosphere whose averaged temperature profile agrees well with the observationally based semi-empirical model of Avrett & Loeser (2008, ApJS, 175, 229).