In this paper, an analysis of the accuracy-enhancement for the discontinuous Galerkin (DG) method applied to one-dimensional scalar nonlinear hyperbolic conservation laws is carried out. This requires analyzing the divided difference of the errors for the DG solution. We therefore first prove that the $$\alpha $$α-th order $$(1 \le \alpha \le {k+1})$$(1≤α≤k+1) divided difference of the DG error in the $$L^2$$L2 norm is of order $${k + \frac{3}{2} - \frac{\alpha }{2}}$$k+32-α2 when upwind fluxes are used, under the condition that $$|f'(u)|$$|f′(u)| possesses a uniform positive lower bound. By the duality argument, we then derive superconvergence results of order $${2k + \frac{3}{2} - \frac{\alpha }{2}}$$2k+32-α2 in the negative-order norm, demonstrating that it is possible to extend the Smoothness-Increasing Accuracy-Conserving filter to nonlinear conservation laws to obtain at least $$({\frac{3}{2}k+1})$$(32k+1)th order superconvergence for post-processed solutions. As a by-product, for variable coefficient hyperbolic equations, we provide an explicit proof for optimal convergence results of order $${k+1}$$k+1 in the $$L^2$$L2 norm for the divided differences of DG errors and thus $$({2k+1})$$(2k+1)th order superconvergence in negative-order norm holds. Numerical experiments are given that confirm the theoretical results.

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非线性标量双曲守恒律的不连续伽辽金方法：划分的差异估计和精度增强

在本文中，对应用于一维标量非线性双曲守恒定律的不连续伽辽金 (DG) 方法的精度增强进行了分析。这需要分析 DG 解决方案的误差划分差异。因此，我们首先证明 $$\alpha $$α 阶 $$(1 \le \alpha \le {k+1})$$(1≤α≤k+1) 除当使用逆风通量时，$$L^2$$L2 范数的顺序为 $${k + \frac{3}{2} - \frac{\alpha }{2}}$$k+32-α2，在 $$|f'(u)|$$|f'(u)| 的条件下 具有统一的正下界。通过对偶性论证，我们然后推导出负阶范数中$${2k + \frac{3}{2} - \frac{\alpha }{2}}$$2k+32-α2 阶的超收敛结果，证明可以将平滑度增加精度守恒滤波器扩展到非线性守恒定律以获得至少 $$({\frac{3}{2}k+1})$$(32k+1) 阶超收敛用于后处理的解决方案。作为副产品，对于可变系数双曲方程，我们为除差的 $$L^2$$L2 范数中 $${k+1}$$k+1 阶的最优收敛结果提供了明确证明DG 误差，因此 $$({2k+1})$$(2k+1)th 阶超收敛在负阶范数中成立。给出的数值实验证实了理论结果。我们为 $$L^2$$L2 范数中 $$L^2$$L2 范数的 $${k+1}$$k+1 阶最优收敛结果提供了明确的证明，因此 $$({2k+1 })$$(2k+1)th 阶超收敛在负阶范数成立。给出的数值实验证实了理论结果。我们为 $$L^2$$L2 范数中 $$L^2$$L2 范数的 $${k+1}$$k+1 阶最优收敛结果提供了明确的证明，因此 $$({2k+1 })$$(2k+1)th 阶超收敛在负阶范数成立。给出的数值实验证实了理论结果。