Abstract: Recent work [doi:10.1016/j.jfa.2005.07.008] has shown that the Degasperis–Procesi equation is well-posed in the class of (discontinuous) entropy solutions. In the present paper we construct numerical schemes and prove that they converge to entropy solutions. Additionally, we provide several numerical examples accentuating that discontinuous (shock) solutions form independently of the smoothness of the initial data. Our focus on discontinuous solutions contrasts notably with the existing literature on the Degasperis–Procesi equation, which seems to emphasize similarities with the Camassa–Holm equation (bi-Hamiltonian structure, integrability, peakon solutions, H¹ as the relevant functional space).