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In this paper, we study the initial-boundary value problem of the Rosenau–Burgers equation by the multiple integral finite volume method (MIFVM). The MIFVM can keep the original equation property very well. We propose a two-level implicit nonlinear discrete scheme, which preserves the energy decline property of the original equation. Existence and uniqueness of the numerical solution are derived. The convergence with the order of O(τ2 + h3) and unconditional stability of the numerical scheme are verified. Numerical examples demonstrate that the scheme is reliable and effective.