Differential-form Xin′anjiang model

The existing Xin′anjiang model is mathematically represented by algebraic equations, and its solution is limited by first-order accuracy finite difference methods, inevitably resulting in numerical errors. Therefore, exploring new approaches to controlling numerical errors is crucial for increasing the computational accuracy of the model. Within the framework of differential systems, the state variables and fluxes of the Xin′anjiang model are identified, and the corresponding control equations and constitutive equations are then derived. A differential-form Xin′anjiang model (ODE-XAJ) is proposed and solved using the fourth-order explicit Runge-Kutta method. The results of the numerical experiments show that, when benchmarked against the analytical solution, the absolute error of ODE-XAJ is on the order of 10^-4 mm or below, enabling a high-order approximation of the analytical solution. With the results of ODE-XAJ as the benchmark, the numerical error of the existing Xin′anjiang model is evaluated to be approximately 8.7% based on the normalized mean absolute error. The application results for a typical watershed show that the determination coefficient of ODE-XAJ increased by 0.02 and the relative error of flood volume decreased by 4.3%. It is concluded that ODE-XAJ theoretically separates the mathematical equations of the model from specific numerical methods, which can effectively control the numerical errors and improve the accuracy of model simulations.