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

TL;DR

This paper analyzes the approximation, convergence, and stability properties of the ADER-DG method for solving ODE systems, establishing its various stability characteristics and practical applicability.

Contribution

It provides a comprehensive theoretical investigation of the ADER-DG method's stability, convergence, and approximation properties, including proofs of key relations for implementation.

Findings

ADER-DG is $A$- and $AN$-stable, $L$-stable, $B$- and $BN$-stable, and algebraically stable.

Theoretical relations for application and implementation are proved.

Applications demonstrate consistency with theoretical results.

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 is $A$- and $AN$-stable, $L$-stable, $B$- and $BN$-stable, and algebraically stable. Several other relations useful for an application and implementation of the ADER-DG method are proved. Applications of the ADER-DG method demonstrated compliance with the expected theoretical results.