One-dimensional secondary settling tank models are widely used in current engineering practice for design and optimization, and usually can be expressed as a nonlinear hyperbolic or nonlinear strongly degenerate parabolic partial differential equation (PDE). Reliable numerical methods are needed to produce approximate solutions that converge to the exact analytical solutions. In this study, we introduced a reliable numerical technique, the Yee–Roe–Davis (YRD) method as the governing PDE solver, and compared its reliability with the prevalent Stenstrom–Vitasovic–Takács (SVT) method by assessing their simulation results at various operating conditions. The YRD method also produced a similar solution to the previously developed Method G and Enquist–Osher method. The YRD and SVT methods were also used for a time-to-failure evaluation, and the results show that the choice of numerical method can greatly impact the solution. Reliable numerical methods, such as the YRD method, are strongly recommended.