This paper presents an effective computing methodology for the Benjamin–Bona–Mahony–Burgers equation with the trigonometric quintic B-spline collocation technique. This methodology employs the trigonometric quintic B-spline collocation method for discretizing spatial derivatives, while the Crank–Nicolson finite difference technique is used for discretizing the temporal derivative. The von Neumann approach has been used to analyze and illustrate that the suggested strategy ensures unconditional stability of the equation. This technique’s efficacy and reliability are shown by numerical examples.