<p>This paper proposes an implicit semi-Lagrangian method relying on a decomposition of operators. Specifically, this paper integrates the Hopmoc algorithm with backward differentiation formulas. Thus, the new strategy employs a multistep approach in time using backward differentiation formulas. This paper discusses both consistency and stability analysis for the novel method when applied to an advection–diffusion equation. The study produces sufficient conditions for consistency analysis and proves that the algorithm presents unconditional stability. Additionally, the numerical results yielded by the approach, when applied to the two-dimensional advection–diffusion equation, confirmed the convergence analysis conducted. The experiments showed that the new hybrid algorithm produces competitive results with several existing methods when applied to the two-dimensional Burgers’ equation.</p>