Video streaming over HTTP/2 is a new trend in multimedia delivery. Compared to the pull-based HTTP/1.1 protocol, the new HTTP/2 protocol's Server Push feature is very effective in reducing the overheads (e.g., in terms of energy, processing, bandwidth) for clients, servers, and network nodes. This paper presents an HTTP/2 push-based adaptation method for on-demand video streaming that reduces the number of requests and provides high video quality. In our method, for each client request, the server sends video segments continuously until it receives another client request with a new quality. Since a request is sent only if the client wants to update the video bitrate, our method can significantly reduce the request related overhead. For this context, a buffer based algorithm is proposed to provide high and smooth video quality while avoiding buffer underflows. Experiments show that the proposed method can provide a lower number of requests, higher average quality and better quality smoothness than existing methods.