In this paper, we propose a family of second and third order temporal integration methods for systems of stiff ordinary differential equations, and explore their application in solving the shallow water equations with friction. The new temporal discretization methods come from a combination of the traditional Runge-Kutta method (for non-stiff equation) and exponential Runge-Kutta method (for stiff equation), and are shown to have both the sign-preserving and steady-state-preserving properties. They are combined with the well-balanced discontinuous Galerkin spatial discretization to solve the nonlinear shallow water equations with non-flat bottom topography and (stiff) friction terms. We have demonstrated that the fully discrete schemes satisfy the well-balanced, positivity-preserving and sign-preserving properties simultaneously. The proposed methods have been tested and validated on several one- and two-dimensional test cases, and good numerical results have been observed.

在本文中，我们提出了一系列用于刚性常微分方程组的二阶和三阶时间积分方法，并探索它们在求解带摩擦的浅水方程中的应用。新的时间离散化方法来自传统的 Runge-Kutta 方法（用于非刚性方程）和指数 Runge-Kutta 方法（用于刚性方程）的组合，并被证明具有保号和稳态-保存属性。它们与平衡良好的不连续 Galerkin 空间离散化相结合，以求解具有非平坦底部地形和（刚性）摩擦项的非线性浅水方程。我们已经证明，完全离散的方案同时满足均衡、正性保留和符号保留的特性。