A novel fourth-order three-point compact operator for the nonlinear convection term uu_x is provided in this paper. The operator makes the numerical analysis of higher-order difference schemes become possible for a wide class of nonlinear evolutionary equations under the unified framework. We take the classical viscous Burgers’ equation as an example and establish a new conservative fourth-order implicit compact difference scheme based on the method of order reduction. A detailed theoretical analysis is carried out by the discrete energy argument and mathematical induction. It is rigorously proved that the difference scheme is conservative, uniquely solvable, stable, and unconditionally convergent in discrete \(L^{\infty }\)-norm. The convergence order is two in time and four in space, respectively. Furthermore, we derive a three-level linearized compact difference scheme for viscous Burgers’ equation based on the proposed operator. All numerically theoretical results similar to that of the nonlinear numerical scheme are inherited completely; meanwhile, it is more time saving. Applying the compact operator to other more complex and higher-order nonlinear evolutionary equations is feasible, including Benjamin-Bona-Mahony-Burgers’ equation, Korteweg-de Vries equation, Kuramoto-Sivashinsky equation, and classification to name a few. Numerical results demonstrate that the presented schemes for Burgers’ equation can achieve second-order accuracy in time and fourth-order accuracy in space in \(L^{\infty }\)-norm.

本文提供了一种新颖的非线性对流项u u _x的四阶三点紧算子。算子使统一框架下的一类非线性演化方程的高阶差分格式的数值分析成为可能。我们以经典粘性Burgers方程为例，建立了一种基于阶约简方法的保守的四阶隐式紧致差分格式。通过离散能量论证和数学归纳法进行了详细的理论分析。严格证明了差分方案在离散\（L ^ {\ infty} \）中是保守的，唯一可解的，稳定的并且无条件地收敛。-规范。收敛顺序在时间上是两个，在空间上是四个。此外，我们基于提出的算子推导了粘性伯格斯方程的三级线性紧致差分格式。所有与非线性数值方案相似的数值理论结果均被完全继承。同时，可以节省更多时间。将紧凑算子应用于其他更复杂和更高阶的非线性演化方程是可行的，包括本杰明-波纳-马奥尼-伯格斯方程，科特维格-德弗里斯方程，仓本-西瓦辛斯基方程以及分类等。数值结果表明，所提出的Burgers方程组在\（L ^ {\ infty} \）-范数下可以达到时间的二阶精度和空间的四阶精度。