<p>In this paper, we consider the efficient solution for an initial-boundary value problem for the Gardner equation: a spatially one-dimensional nonlinear evolution equation describing a broad class of dispersive autowave processes. We propose a numerical-analytical method based on a combination of explicit and implicit time discretization schemes for various terms of the differential operator. We develop a new efficient algorithm to solve a sequence of auxiliary linear problems, relying on analytical representations using an explicit form of the fundamental system of solutions. We consider an example of a numerical solution of the initial-boundary value problem for the Gardner equation and compare the result with a known exact solution of the solitary traveling wave type.</p>