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

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.