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

Abstract: 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.