非线性双曲方程的间断有限元超收敛分析和应用

This project is dedicated to the superconvergence properties and its application of the discontinuous Galerkin method for the nonlinear hyperbolic equations which are widely used in various physical fields. However, convergence and superconvergence analysis of numerical methods for these equations is difficult, due to the nonlinear term. Based on the existed discontinuous Galerkin methods for nonlinear hyperbolic equations, high order numerical approximation and superconvergence properties of these methods will be discussed and studied. The superconvergence results furthermore provide better understanding of the advantage of the numerical method, and theoretical basis to a posteriori error estimate, adaptive algorithm and post-processing. The correction functions idea, Taylor expansion and a prior error assumption will be used in this projection to study and analyze the superconvergence properties: on one hand, we construct a specially designed correction function to correct the error, and choose suitable initial and boundary discretizations to obtain high order error; on the other hand, we used the Taylor expansion and a prior error assumption to handle the nonlinearity. By doing so, the superconvergence analysis for nonlinear problems is reduced to the superconvergence for the linear case and the estimate for a high order residual. Then superconvergence results of the discontinuous Galerkin approximation are obtained for the smooth solution or the smooth region away from the discontinuity.

本项目研究非线性双曲方程的间断有限元超收敛理论及其应用。非线性双曲方程具有广泛的物理背景和很强的应用价值。然而受非线性项的影响，研究该类方程的数值方法的收敛乃至超收敛性具有较大困难。在已有的求解非线性双曲方程的间断有限元方法的基础上，本项目将进一步研究这类数值方法的高精度逼近和超收敛性质，进而更深层地揭示数值方法的优点，为后验误差估计、自适应算法、后处理恢复技术等各科学计算领域提供重要理论基础。本项目将采用校正函数、泰勒展开和先验假设等方式来开展超收敛研究：一方面通过设计特殊的校正函数，选择新的初值和边值离散化方法，利用消除技巧获取高阶精度，另一方面采用泰勒展开和先验假设的技巧来处理非线性项，把非线性问题的间断有限元超收敛研究分解为线性问题的超收敛分析和含高精度的余项估计，以此获得间断有限元逼近在光滑解或远离解的间断处的超收敛结果。

本项目研究非线性双曲方程的间断有限元超收敛理论及其应用。鉴于非线性问题本身的复杂性，本项目的前期工作主要围绕线性问题的间断有限元超收敛性质开展研究，其中包含偏迎风间断有限元法的超收敛性以及线性退化变系数的双曲守恒律问题、含高阶导数的线性偏微分方程等的间断有限元法的超收敛研究。这些超收敛理论的研究，为本项目进一步分析非线性双曲问题提供了非常重要的理论基础和技术依托。在此基础上，通过构造特殊的校正函数，选择新的初值和边值离散化方法，采用泰勒展开和先验假设等处理非线性项的方式，本项目获得了间断有限元逼近在光滑解或远离解的间断处的超收敛结果，得到了关于非线性问题在风向不变和风向变化情况下两类截然不同的超收敛结果。这些超收敛结果将为后验误差估计、自适应算法等领域提供理论指导。