Burgers 方程是广泛应用于众多领域的一类重要的非线性发展方程，研究其高性能数值解法具有重要的理论意义和应用价值。数值求解Burgers方程，需要处理离散化后的非线性代数系统，对大规模、高精度问题而言，将会遇到存储量大、计算复杂度高的瓶颈问题。针对这一困难，本项目拟采用多尺度Galerkin方法来离散化Burgers方程的空间变量，在每个时间层上，提出快速求解离散化所得非线性方程组的多层扩充算法，预期算法以准线性的计算复杂度得到最优收敛阶。在此基础上，针对高雷诺数Burgers方程的解局部出现激波的特性，利用多尺度基底对函数的极强表现力，并充分挖掘多尺度离散所得代数系统的多尺度层次性，研究将空间变量自适应多尺度分解与多层扩充法相结合的快速算法，以提高求解的稳定性和效率。研究成果将进一步丰富多尺度数值计算方法的理论，并为其它类型的非线性发展方程的数值计算提供借鉴。

The Burgers equation is an important evolutional equation which is applied in a lot of fields. To develop high performance numerical methods for solving the equation is of important theoretical significance and great application value. Its numerical solutions normally require solving the nonlinear algebraic system generated by discretization. When high approximation accuracy is desired, it requires one to use a sufficiently small time-step and sufficiently fine grids. Thus it demands a large amount of storage space and computational effort. This becomes a bottleneck problem for numerical solutions of the Burgers equation. To overcome this difficulty, we will use the multiscale Galerkin method to discrete the spacial variable. At each time step, we will develop the multilevel augmentation method for fast solving the corresponding nonlinear systems, and give the theoretical analysis for the convergence of the method. The proposed algrothim will obtain the optimal oder of convergence with quasi-linear computational complexity. Another difficulty for numerically solving the Burgers equation is that in the case of high Reynolds number local shock waves may appear in the solution, which may affect the accuracy. We will combine the adaptive multiscale decomposition and the multilevel augmentation method to deal with this problem. In the procedure of solving the equation, we will take full advantage of the strong expression of the multiscale basis and thoroughly dig out the multilevel property of the resulting algebraic system to construct more stable and efficient algorithms. The researches of this project will enrich the theory of multiscale methods, and provide a reference for numerically solving other nonlinear evolutional equations.