PiTSBiCG: Parallel in Time Stable Bi-Conjugate Gradient Algorithm

TL;DR

The paper introduces PiTSBiCG, a novel parallel in time algorithm using a stabilized bi-conjugate gradient method, offering improved stability and performance over existing methods for solving time-dependent differential equations.

Contribution

It presents a new algebraic formulation and algorithm for parallel in time simulation, enhancing stability and efficiency compared to the parareal method.

Findings

PiTSBiCG demonstrates superior stability over parareal.

Numerical experiments confirm improved convergence.

Algorithm effectively handles various differential equations.

Abstract

This paper presents a new algorithm for the parallel in time (PiT) numerical simulation of time dependent partial/ordinary differential equations. We propose a reliable alternative to the well know parareal in time algorithm, by formulating the parallel in time problem algebraically and solve it using an adapted Bi-Conjugate gradient stabilized method. The proposed Parallel in time Stable Bi-Conjugate algorithm (PiTSBiCG) has a great potential in stabilizing the parallel resolution for a variety of problems. In this work, we describe the mathematical approach to the new algorithm and provide numerical evidences that show its superiority to the standard parareal method.