The numerical error of the Xinanjiang model

The numerical error of the Xinanjiang (XAJ) model has long been ignored, despite its wide application in China. Separating the mathematically formulated model from the computationally formulated model and further solving by an appropriate time-stepping scheme provides a chance to systematically examine the numerical error of the XAJ model because the exact analytical solution is difficult to obtain owing to its nonlinearity. The mathematically formulated model is proposed by deriving a state-space representation of the XAJ model by identifying the state variables, fluxes, and governing differential equations. Five experiments were conducted around the original XAJ model and its state-space representation with different parameter groups, input series, and time-stepping schemes to examine the numerical error at the submodule and model levels. We found that only the tension water storage capacity curve submodule of the original XAJ model is numerical error-free, and the adaptive-step explicit fourth-order Runge-Kutta scheme is the most efficient because it balances the accuracy, efficiency, and physical constraint preservation. The numerical error of the total amount of the discharge at the watershed outlet within a time period (Q∗) produced by the original XAJ model is relatively large, which mainly comes from the free water storage capacity curve submodule. The large numerical error of Q∗ occurs with high precipitation intensity, and decreases as the flood receded. The submodules related to the runoff generation process introduce and amplify numerical errors, while the submodules related to the runoff concentration process has the ability to alleviate numerical errors. This study helps to deepen our understanding of the numerical error of the XAJ model, and thus guides its application and modification.