Rumours are a kind of information that has a impact on social life and economies, which spread quickly and widely, especially, via internet. Recently, the spread of information affects our daily lives due to the increasing number of social media users. Rumours are defined in social areas which are delivered by gossips, fake news, marketing, social media all the way even to revolutions. In this paper, we study the dynamics of a rumour propagation model with a numerical approach. By using an algorithmic technique with an error analysis, the validity of the numerical technique is described. We investigate the model with this numerical approach to explain the dynamics of rumour propagation. Besides, we explain sensitivity analyses of the model of parameters. Then by numerical simulations efficiency of the technique is shown. Finally, the results are displayed and discussed with the help of figures and tables. The paper ends with a conclusion and an outlook to future studies.