标题： 任意高阶 ADER-DG 方法与局部 DG 预测器用于一阶常微分方程组初值问题的解 显示英文标题

标题： Arbitrary high order ADER-DG method with local DG predictor for solutions of initial value problems for systems of first-order ordinary differential equations

摘要： 对用于求解一阶非线性ODE系统的IVP的任意高阶ADER-DG数值方法进行适应，引入了局部DG预测器。所提出的数值方法是一个完全的单步ODE求解器，具有统一的步长，并且在算法和软件实现上都很简单。已经证明，所提出的ADER-DG数值方法是A稳定的和L稳定的。ADER-DG数值方法在网格节点处的解具有2N+1的超收敛阶数，而使用局部DG预测器获得的局部解具有N+1的收敛阶数。已经证明，该数值方法实现的一个重要应用特性是能够将局部解用作具有亚网格分辨率的解，这使得即使在非常粗糙的坐标网格上也可以获得详细的解。当使用标准的单精度或双精度浮点数表示进行计算，并使用较大的N值时，局部解的误差范围几乎与网格节点处解的误差没有区别。展示了ADER-DG方法在求解由极端刚度特征的刚性ODE系统方面的能力。获得了ADER-DG数值方法的计算成本估计。 显示英文摘要

摘要： An adaptation of the arbitrary high order ADER-DG numerical method with local DG predictor for solving the IVP for a first-order non-linear ODE system is proposed. The proposed numerical method is a completely one-step ODE solver with uniform steps, and is simple in algorithmic and software implementations. It was shown that the proposed version of the ADER-DG numerical method is A-stable and L-stable. The ADER-DG numerical method demonstrates superconvergence with convergence order 2N+1 for the solution at grid nodes, while the local solution obtained using the local DG predictor has convergence order N+1. It was demonstrated that an important applied feature of this implementation of the numerical method is the possibility of using the local solution as a solution with a subgrid resolution, which makes it possible to obtain a detailed solution even on very coarse coordinate grids. The scale of the error of the local solution, when calculating using standard representations of single or double precision floating point numbers, using large values of the degree N, practically does not differ from the error of the solution at the grid nodes. The capabilities of the ADER-DG method for solving stiff ODE systems characterized by extreme stiffness are demonstrated. Estimates of the computational costs of the ADER-DG numerical method are obtained.