This article develops a new discontinuous Galerkin (DG) method with the one-stage arbitrary derivatives in time and space approach to solve one-dimensional hyperbolic conservation laws. This method employs the differential transformation procedure instead of the Cauchy–Kowalewski procedure to recursively express the spatiotemporal expansion coefficients of the solution through the low-order spatial expansion coefficients. The proposed method is free of solving generalized Riemann problems at inter-cells. Compared with the Runge–Kutta DG methods, the current method needs less computer memory due to no intermediate stages. In summary, this method is one step, one stage, fully discrete, and easily achieves arbitrary high-order accuracy in time and space. Extensive numerical results illustrate the good performances of the present method: high-order accuracy for smooth solutions, good resolution for discontinuous solutions and high efficiency.

本文开发了一种新的不连续伽勒金（DG）方法，该方法采用时空一阶任意导数的方法来求解一维双曲守恒律。该方法采用微分变换过程而不是Cauchy-Kowalewski过程来通过低阶空间扩展系数来递归地表示解的时空扩展系数。所提出的方法没有解决小区间的广义黎曼问题。与Runge–Kutta DG方法相比，由于没有中间阶段，当前方法需要较少的计算机内存。总之，该方法是一个步骤，一个阶段，完全离散，并且可以轻松地在时间和空间上实现任意的高阶精度。大量的数值结果说明了本方法的良好性能：