A numerical method ADER-DG with a local DG predictor for solving a DAE system has been developed, which was based on the formulation of ADER-DG methods using a local DG predictor for solving ODE and PDE systems. The basis functions were chosen in the form of Lagrange interpolation polynomials with nodal points at the roots of the Radau polynomials, which differs from the classical formulations of the ADER-DG method, where it is customary to use the roots of Legendre polynomials. It was shown that the use of this basis leads to A-stability and L1-stability in the case of using the DAE solver as ODE solver. The numerical method ADER-DG allows one to obtain a highly accurate numerical solution even on very coarse grids, with a step greater than the main characteristic scale of solution variation. The local discrete time solution can be used as a numerical solution of the DAE system between grid nodes, thereby providing subgrid resolution even in the case of very coarse grids. The classical test examples were solved by developed numerical method ADER-DG. With increasing index of the DAE system, a decrease in the empirical convergence orders p is observed. An unexpected result was obtained in the numerical solution of the stiff DAE system -- the empirical convergence orders of the numerical solution obtained using the developed method turned out to be significantly higher than the values expected for this method in the case of stiff problems. It turns out that the use of Lagrange interpolation polynomials with nodal points at the roots of the Radau polynomials is much better suited for solving stiff problems. Estimates showed that the computational costs of the ADER-DG method are approximately comparable to the computational costs of implicit Runge-Kutta methods used to solve DAE systems. Methods were proposed to reduce the computational costs of the ADER-DG method.

翻译：本文发展了一种结合局部DG预测器的ADER-DG数值方法用于求解DAE系统，该方法基于采用局部DG预测器求解ODE和PDE系统的ADER-DG方法框架。基函数选用以Radau多项式根为节点的拉格朗日插值多项式形式，这与经典ADER-DG方法通常采用Legendre多项式根的惯例不同。研究表明，当将DAE求解器作为ODE求解器使用时，采用该基函数可保证A-稳定性和L1-稳定性。该ADER-DG数值方法即使在步长大于解变化主特征尺度的极粗网格上，仍能获得高精度数值解。局部离散时间解可用作网格节点间DAE系统的数值解，从而在极粗网格情况下仍能提供亚网格分辨率。通过经典算例验证了所发展的ADER-DG数值方法。随着DAE系统指标的增加，观察到经验收敛阶数p的下降。在刚性DAE系统的数值求解中获得了意外结果——采用该方法所得数值解的经验收敛阶数显著高于该方法在刚性问题上预期值。结果表明，采用以Radau多项式根为节点的拉格朗日插值多项式更适合求解刚性问题。评估显示ADER-DG方法的计算成本与用于求解DAE系统的隐式Runge-Kutta方法大致相当。本文还提出了降低ADER-DG方法计算成本的若干途径。