Dirac方程是量子力学中的重要方程，在二维材料、凝聚态物理等科学领域也有广泛的应用，为其设计高效稳定的数值算法是科学研究与工程实践中的前沿热点问题。在非相对论极限下，方程的解在时间上具有高振荡性，这为设计高效准确的数值算法带来了巨大的挑战。传统算法需要时间步长小于振荡波长才能保证算法的准确性，文献中的多尺度算法虽然能够摆脱网格尺度的限制实现一致准确性，但在不增加计算量的前提下一般只能达到一阶一致收敛。本项目针对非相对论极限下的Dirac方程，拟采用多尺度分析方法、Picard迭代法、指数积分因子法、谱方法等，为其设计高效高阶一致准确的算法，并建立算法的误差理论。本项目的研究成果将为应用中所需的高效高精度数值模拟提供必要的理论和技术支持。

Dirac equation is an important equation in quantum mechanics, and widely used in many other fields, such as two dimensional materials and condensed physics. Therefore, it has long been of great importance to design stable and efficient numerical methods for solving the Dirac equation. In the non-relativistic regime, the solution to the Dirac equation is highly oscillatory in time. This kind of temporal oscillation makes it quite challenging for developing efficient and accurate numerical methods. In order to capture the correct solution, traditional methods usually suffer from severe time step restrictions, i.e., the time step is required to be smaller than the wave length of the oscillation. Multiscale methods are able to get rid of the meshing strategy restriction and achieve uniform accuracies. However, without loss of the efficiency, the methods in the literature could only achieve a uniform linear convergence rate. In this project, we will propose efficient and high-order uniformly accurate methods for the Dirac equation in the non-relativistic regime, by utilizing multiscale analysis, Picard iteration, exponential wave integrators and spectral methods. We also rigorously establish the error estimates for the proposed method. The study of this project will provide theoretical and technical supports for efficient simulations in applications.