Unified Analysis for Variational Time Discretizations of Higher Order and Higher Regularity Applied to Non-stiff ODEs

TL;DR

This paper provides a comprehensive analysis of variational time discretization methods, including discontinuous and continuous Galerkin approaches, for non-stiff ODEs, establishing error bounds, superconvergence, and supporting numerical validation.

Contribution

It offers a unified theoretical framework for analyzing various Galerkin-based time discretization methods with weak assumptions and broad applicability.

Findings

Global error estimates are established for the methods.

Superconvergence properties are demonstrated.

Numerical experiments confirm theoretical predictions.

Abstract

We present a unified analysis for a family of variational time discretization methods, including discontinuous Galerkin methods and continuous Galerkin-Petrov methods, applied to non-stiff initial value problems. Besides the well-definedness of the methods, the global error and superconvergence properties are analyzed under rather weak abstract assumptions which also allow considerations of a wide variety of quadrature formulas. Numerical experiments illustrate and support the theoretical results.