Delay Sobolev equations (DSEs) are a class of important models in fluid mechanics, thermodynamics and the other related fields. For solving this class of equations, in this paper, linearized compact difference methods (LCDMs) for one- and two-dimensional problems of DSEs are suggested. The solvability and convergence of the methods are analyzed and it is proved under some appropriate conditions that the methods are convergent of order two in time and order four in space. In order to improve the computational accuracy of LCDMs in time, we introduce the Richardson extrapolation technique, which leads to the improved LCDMs can reach the fourth-order accuracy in both time and space. Finally, with several numerical experiments, the theoretical accuracy and computational effectiveness of the proposed methods are further testified.

延迟Sobolev方程（DSE）是流体力学，热力学和其他相关领域中的一类重要模型。为了解决此类方程，在本文中，针对DSE的一维和二维问题，提出了线性化的紧致差分方法（LCDM）。分析了该方法的可解性和收敛性，并在一定条件下证明了该方法在时间上是二阶且在空间上是四阶收敛。为了及时提高LCDMs的计算精度，我们引入了Richardson外推技术，从而使改进后的LCDMs在时间和空间上都能达到四阶精度。最后，通过几个数值实验，进一步验证了所提方法的理论准确性和计算有效性。