Abstract: A posteriori $L^1$ error estimates (in the sense of [12, 23]) are derived for both well-balanced (WB) and fractional-step (FS) numerical approximations of the unique weak solution of the Cauchy problem for the 1D semilinear damped wave equation. For setting up the WB algorithm, we proceed by rewriting it under the form of an elementary 2 × 2 system which linear convective structure allows to reduce the Godunov scheme with optimal Courant number (corresponding to $∆t = ∆x$) to a wavefront-tracking algorithm free from any step of projection onto piecewise constant functions. A fundamental difference in the total variation estimates is proved, which partly explains the discrepancy of the FS method when the dissipative (sink) term displays an explicit dependence in the space variable. Numerical tests are performed by means of several exact solutions of the linear damped wave equation.