标题： ADER-DG方法在常微分方程中的理论和内部结构 显示英文标题

标题： Theory and internal structure of ADER-DG method for ordinary differential equations

摘要： 对求解ODE系统的ADER-DG方法的近似性质、收敛性和稳定性进行了研究。 ADER-DG方法生成一个新的隐式RK方法，其性质与原始ADER-DG方法相似。 当使用次数为$N$的多项式进行网格节点上的数值解时，ADER-DG方法具有近似阶$2N+1$，并表现出超收敛。 由局部DG预测器得到的局部解具有近似阶$N+1$，并具有亚网格分辨率。 该ADER-DG方法是$A$-和$AN$-稳定的，$L$-稳定的，$B$-和$BN$-稳定的，并且代数稳定。 ADER-DG方法的应用展示了与预期理论结果的一致性。 显示英文摘要

摘要： Investigation of the approximation properties, convergence, and stability of the ADER-DG method for solving an ODE system is carried out. The ADER-DG method generates a new implicit RK method, which is similar in its properties to the original ADER-DG method. The ADER-DG method has an approximation order $2N+1$ when using polynomials of degree $N$ for the numerical solution at grid nodes, and demonstrates superconvergence. The local solution obtained by the local DG predictor has an approximation order $N+1$ and has a subgrid resolution. The ADER-DG method is $A$- and $AN$-stable, $L$-stable, $B$- and $BN$-stable, and algebraically stable. Applications of the ADER-DG method demonstrated compliance with the expected theoretical results.