Arbitrary high order ADER-DG method with local DG predictor for solutions of initial value problems for systems of first-order ordinary differential equations

TL;DR

This paper introduces an adapted high-order ADER-DG numerical method with a local DG predictor for solving initial value problems of first-order nonlinear ODE systems, emphasizing stability, superconvergence, and detailed solutions on coarse grids.

Contribution

It presents a novel, simple, one-step ADER-DG method with local predictors that is A-stable, L-stable, and capable of high-order superconvergent solutions, suitable for stiff ODEs and coarse grids.

Findings

The method is A-stable and L-stable.

Achieves superconvergence with order 2N+1 at grid nodes.

Enables detailed solutions on coarse grids with minimal error increase.

Abstract

An adaptation of the arbitrary high order ADER-DG numerical method with local DG predictor for solving the IVP for a first-order non-linear ODE system is proposed. The proposed numerical method is a completely one-step ODE solver with uniform steps, and is simple in algorithmic and software implementations. It was shown that the proposed version of the ADER-DG numerical method is A-stable and L-stable. The ADER-DG numerical method demonstrates superconvergence with convergence order 2N+1 for the solution at grid nodes, while the local solution obtained using the local DG predictor has convergence order N+1. It was demonstrated that an important applied feature of this implementation of the numerical method is the possibility of using the local solution as a solution with a subgrid resolution, which makes it possible to obtain a detailed solution even on very coarse coordinate grids. The scale of the error of the local solution, when calculating using standard representations of single or double precision floating point numbers, using large values of the degree N, practically does not differ from the error of the solution at the grid nodes. The capabilities of the ADER-DG method for solving stiff ODE systems characterized by extreme stiffness are demonstrated. Estimates of the computational costs of the ADER-DG numerical method are obtained.