# A priori error estimates of Adams-Bashforth discontinuous Galerkin methods for scalar nonlinear conservation laws

## TL;DR

This paper establishes theoretical convergence and error estimates for Adams-Bashforth and forward Euler discontinuous Galerkin methods applied to scalar nonlinear conservation laws, highlighting optimal temporal rates and suboptimal spatial rates under CFL conditions.

## Contribution

It provides the first a priori error estimates for Adams-Bashforth DG methods and extends convergence analysis to these time-stepping schemes for nonlinear conservation laws.

## Key findings

- Optimal convergence rates in time for the Adams-Bashforth method
- Suboptimal convergence rates in space for the DG methods
- Error estimates valid under CFL conditions

## Abstract

In this paper we show theoretical convergence of a second-order Adams-Bashforth discontinuous Galerkin method for approximating smooth solutions to scalar nonlinear conservation laws with E-fluxes. A priori error estimates are also derived for a first-order forward Euler discontinuous Galerkin method. Rates are optimal in time and suboptimal in space; they are valid under a CFL condition.