Stable parareal in time method for first and second order hyperbolic system

TL;DR

This paper introduces a stable variant of the parareal in time algorithm tailored for hyperbolic systems, improving stability and accuracy in long-term simulations of wave and conservation law problems.

Contribution

The paper proposes a new stable parareal in time method specifically designed for hyperbolic systems, addressing stability issues and extending its applicability.

Findings

The new method successfully solves linear wave equations.

It effectively handles nonlinear Burger's equations.

Results demonstrate improved stability over the plain parareal in time algorithm.

Abstract

The parareal in time algorithm allows to perform parallel simulations of time dependent problems. This algorithm has been implemented on many types of time dependent problems with some success. Recent contributions have allowed to extend the domain of application of the parareal in time algorithm so as to handle long time simulations of Hamiltonian systems. This improvement has managed to avoid the fatal large lack of accuracy of the plain parareal in time algorithm consequence of the fact that the plain parareal in time algorithm does not conserve invariants. A somehow similar difficulty occurs for problems where the solution lacks regularity, either initially or in the evolution, like for the solution to hyperbolic system of conservation laws. In this paper we identify the problem of lack of stability of the parareal in time algorithm and propose a simple way to cure it. The new method is used to solve a linear wave equation and a non linear Burger's equation, the results illustrate the stability of this variant of the parareal in time algorithm.