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This is a Experimental Study study.

October 11, 2025Open Access

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

Key Points

The ADER-DG method achieves an approximation order of 2N+1 using polynomials of degree N for numerical solutions.
It demonstrates superconvergence and employs a local DG predictor with an approximation order of N+1.
The method exhibits multiple forms of stability, including A-, AN-, L-, B-, and BN-stability, maintaining algebraic stability.
Applications align with theoretical expectations, confirming the robustness of the ADER-DG method in practice.

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.

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Cite This Study

И. С. Попов (Tue,) studied this question.

synapsesocial.com/papers/68e9b2e4ba7d64b6fc133191 https://doi.org/https://doi.org/10.48550/arxiv.2508.13824