The impact of numerical error on hydrological model performance

The simple yet crucial question of what impact numerical error has on performance of hydrological model has not received consistent answers. Discrepancies in previous studies can be attributed to model structural changes and insufficient watershed sample sizes. In this study, we focus on the Xinanjiang model, a widely used hydrological model in China. Two mathematical forms are considered: the original discrete equation form (DTE-XAJ), which prone to numerical error, and the differential equation form (ODE-XAJ), which serves as an reference with controlled numerical error. To assess structure consistency, we propose a convergence validation method by gradually shorting time step of the DTE-XAJ model and checking for convergence to numerical solution of the ODE-XAJ model. To address sample size limitations, we apply both models to 547 watersheds from the CAMELS dataset. The results show that: (1) state variables and fluxes of the DTE-XAJ model converge to those of the ODE-XAJ model under time step refinement, demonstrating structural consistency and the effectiveness of the convergence validation method; (2) numerical error significantly degrades Xinanjiang model performance during both calibration and validation period, meaning its negative effects cannot be fully compensated by the calibration procedure; (3) controlling numerical error leads to systematic performance improvements even for a mature model such as XAJ, providing empirical evidence of its non-negligible contribution to model uncertainty. This study proposes the convergence validation method to rigorously assess structural consistency, providing a theoretical foundation for future research. The findings underscore the importance of addressing numerical error in hydrological models.