In this paper, we study superconvergence properties of the local discontinuous Galerkin (LDG) methods for solving nonlinear convection-diffusion equations in one space dimension. The main technicality is an elaborate estimate to terms involving projection errors. By introducing a new projection and constructing some correction functions, we prove the \((2k+1)\)th order superconvergence for the cell averages and the numerical flux in the discrete \(L^2\) norm with polynomials of degree \(k\ge 1\), no matter whether the flow direction \(f'(u)\) changes or not. Superconvergence of order \(k +2\) (\(k +1\)) is obtained for the LDG error (its derivative) at interior right (left) Radau points, and the convergence order for the error derivative at Radau points can be improved to \(k+2\) when the direction of the flow doesn’t change. Finally, a supercloseness result of order \(k+2\) towards a special Gauss–Radau projection of the exact solution is shown. The superconvergence analysis can be extended to the generalized numerical fluxes and the mixed boundary conditions. All theoretical findings are confirmed by numerical experiments.

在本文中，我们研究了求解一维非线性对流扩散方程的局部不连续Galerkin（LDG）方法的超收敛性质。主要技术是对涉及投影误差的术语进行详尽的估计。通过引入一个新的投影并构造一些校正函数，我们证明了单元平均的\（（2k + 1）\）阶超收敛性和多项式为\的离散\（L ^ 2 \）范数中的数值通量。 （k \ ge 1 \），无论流向\（f'（u）\）是否改变。阶\（k +2 \）（\（k +1 \）的超收敛）是针对内部右（左）Radau点处的LDG误差（其导数）获得的，并且当流向不正确时，可以将Radau点处的误差导数的收敛阶提高为\（k + 2 \）。改变。最后，显示了对精确解的特殊高斯-拉多投影阶为\（k + 2 \）的超闭合性结果。超收敛分析可以扩展到广义数值通量和混合边界条件。所有的理论发现均通过数值实验得到证实。