In this paper, we study an ultra-weak local discontinuous Galerkin (UWLDG) method for the KdV–Burgers–Kuramoto (KBK) type equation. While the standard UWLDG method is a powerful tool for efficiently solving high order equations, it faces challenges when applied to equations involving multiple spatial derivatives. We adopt a novel approach to discretize lower order spatial derivatives, enhancing the versatility of the UWLDG method. Additionally, we adopt generalized numerical fluxes to enhance the flexibility and extendibility of the UWLDG scheme. We introduce a class of global projections with multiple parameters to analyze the properties of these generalized numerical fluxes. With the aid of the special discretization approach and the global projections, we establish both stability and optimal error estimates of proposed method. The validity of our theoretical findings is demonstrated through numerical experiments.

在本文中，我们研究了 KdV-Burgers-Kuramoto (KBK) 型方程的超弱局部不连续 Galerkin (UWLDG) 方法。虽然标准 UWLDG 方法是有效求解高阶方程的强大工具，但在应用于涉及多个空间导数的方程时面临挑战。我们采用一种新颖的方法来离散低阶空间导数，增强了 UWLDG 方法的通用性。此外，我们采用广义数值通量来增强 UWLDG 方案的灵活性和可扩展性。我们引入一类具有多个参数的全局投影来分析这些广义数值通量的属性。借助特殊的离散化方法和全局投影，我们建立了所提出方法的稳定性和最优误差估计。我们的理论研究结果的有效性通过数值实验得到了证明。