Improving the local solution of the DG predictor of the ADER-DG method for solving systems of ordinary differential equations and its applicability to systems of differential-algebraic equations

TL;DR

This paper introduces an improved local numerical solution for the ADER-DG method that enhances accuracy, smoothness, and convergence order without significant structural changes, and extends applicability to differential-algebraic equations.

Contribution

The paper develops a higher-order, continuous local numerical solution for ADER-DG, with rigorous proofs and broad application to ODE and DAE systems, maintaining stability and superconvergence.

Findings

Convergence order of the improved solution exceeds the original by one.

The improved solution exhibits higher accuracy and smoothness.

Empirical results confirm theoretical convergence orders.

Abstract

Improved local numerical solution for the ADER-DG numerical method with a local DG predictor for solving the initial value problem for a first-order ODE system is proposed. The improved local numerical solution demonstrates convergence orders of one higher than the convergence order of the local numerical solution of the original ADER-DG numerical method and has the property of continuity at grid nodes. Rigorous proofs of the approximation orders of the local numerical solution and the improved local numerical solution are presented. Obtaining the proposed improved local numerical solution does not require significant changes to the structure of the ADER-DG numerical method. Therefore, all conclusions regarding the convergence orders of the numerical solution at grid nodes, the resulting superconvergence, and the high stability of the ADER-DG numerical method remain unchanged. A wide range of applications of the ADER-DG numerical method is presented for solving specific initial value problems for ODE systems for a wide range of polynomial degrees. The obtained results provide strong confirmation for the developed rigorous theory. The improved local numerical solution is shown to exhibit both higher accuracy and improved smoothness and point-wise comparability. Empirical convergence orders of all individual numerical solutions were calculated for a wide range of error norms, which well agree with the expected convergence orders. The rigorous proof, based on the $\epsilon$-embedding method, of the applicability of the ADER-DG numerical method with a local DG predictor to solving DAE systems is presents.