An Iterative Approach for Time Integration Based on Discontinuous Galerkin Methods

TL;DR

This paper introduces a new iterative framework for time integration using discontinuous Galerkin methods, enabling high-order, stable, and efficient schemes suitable for parallel and multilevel computations in solving ODEs and PDEs.

Contribution

It develops a systematic approach to construct explicit, implicit, and semi-implicit high-order schemes based on DG, with proven accuracy and stability, and integrates multilevel strategies for accelerated convergence.

Findings

Implicit schemes are at least A-stable for p ≤ 9.

The schemes achieve up to 2p+1 order accuracy.

Multilevel strategies significantly improve convergence speed.

Abstract

We present a new class of iterative schemes for solving initial value problems (IVP) based on discontinuous Galerkin (DG) methods. Starting from the weak DG formulation of an IVP, we derive a new iterative method based on a preconditioned Picard iteration. Using this approach, we can systematically construct explicit, implicit and semi-implicit schemes with arbitrary order of accuracy. We also show that the same schemes can be constructed by solving a series of correction equations based on the DG weak formulation. The accuracy of the schemes is proven to be $\min\{2p+1, K+1\}$ with $p$ the degree of the DG polynomial basis and $K$ the number of iterations. The stability is explored numerically; we show that the implicit schemes are $A$-stable at least for $0 \leq p \leq 9$. Furthermore, we combine the methods with a multilevel strategy to accelerate their convergence speed. The new multilevel scheme is intended to provide a flexible framework for high order space-time discretizations and to be coupled with space-time multigrid techniques for solving partial differential equations (PDEs). We present numerical examples for ODEs and PDEs to analyze the performance of the new methods. Moreover, the newly proposed class of methods, due to its structure, is also a competitive and promising candidate for parallel in time algorithms such as Parareal, PFASST, multigrid in time, etc.