微分形式新安江模型

Differential-form Xin′anjiang model

现有新安江模型数学上是代数方程并限于一阶差分方法求解,不可避免存在数值误差,探索数值求解误差控制新途径,对于提高模型计算精度具有重要意义。基于微分系统框架,识别新安江模型的状态变量和通量,推导其控制方程和本构方程,提出微分形式的新安江模型(ODE-XAJ),并采用四阶显式Runge Kutta法求解。数值实验结果显示,以解析解为基准,ODE-XAJ绝对误差处于或小于10^(-4)mm量级,可实现对解析解的高阶近似;以ODE-XAJ结果为基准,按归一化平均绝对误差评价,现有新安江模型的数值求解误差约为8.7%。典型流域应用结果显示,ODE-XAJ确定性系数提升0.02,洪量相对误差降低4.3%。研究表明,ODE-XAJ理论上分离了模型的数学方程与具体解法,可有效控制数值求解误差,提升模型模拟精度。

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.