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Despite the growing interest in parallel-in-time methods as an approach to accelerate numerical simulations in atmospheric modelling, improving their stability and convergence remains a substantial challenge for their application to…

Numerical Analysis · Mathematics 2023-10-27 João Guilherme Caldas Steinstraesser , Pedro da Silva Peixoto , Martin Schreiber

The Parareal algorithm is used to solve time-dependent problems considering multiple solvers that may work in parallel. The key feature is a initial rough approximation of the solution that is iteratively refined by the parallel solvers. We…

Systems and Control · Computer Science 2014-02-18 Loïc Michel

In a variety of scientific and engineering domains, the need for high-fidelity and efficient solutions for high-frequency wave propagation holds great significance. Recent advances in wave modeling use sufficiently accurate fine solver…

Analysis of PDEs · Mathematics 2025-09-17 Luis Kaiser , Richard Tsai , Christian Klingenberg

Simulating large-scale microswimmer dynamics in viscous fluid poses significant challenges due to the coupled high spatial and temporal complexity. Conventional high-performance computing (HPC) methods often address these two dimensions in…

Distributed, Parallel, and Cluster Computing · Computer Science 2026-04-15 Ruixiang Huang , Weifan Liu

In this paper, we propose a model order reduction based adaptive parareal method for time-dependent partial differential equations. By using the data obtained by the fine propagator in each iteration of the plain parareal method together…

Numerical Analysis · Mathematics 2026-01-19 Xiaoying Dai , Miao Hu , Shuwei Shen

We present a parareal in time algorithm for the simulation of neutron diffusion transient model. The method is made efficient by means of a coarse solver defined with large time steps and steady control rods model. Using finite element for…

While recent advances in deep learning have shown promising efficiency gains in solving time-dependent partial differential equations (PDEs), matching the accuracy of conventional numerical solvers still remains a challenge. One strategy to…

Numerical Analysis · Mathematics 2025-11-26 Yuwei Geng , Junqi Yin , Eric C. Cyr , Guannan Zhang , Lili Ju

A variant of the Parareal method for highly oscillatory systems of PDEs was proposed by Haut and Wingate (2014). In that work they proved superlinear conver- gence of the method in the limit of infinite time scale separation. Their coarse…

Numerical Analysis · Mathematics 2017-05-29 Adam Peddle , Terry Haut , Beth Wingate

This paper presents a highly-parallelizable parallel-in-time algorithm for efficient solution of nonlinear time-periodic problems. It is based on the time-periodic extension of the Parareal method, known to accelerate sequential…

Numerical Analysis · Mathematics 2020-07-08 Iryna Kulchytska-Ruchka , Sebastian Schöps

A high-performance parallel algorithm is proposed for modeling the propagation of acoustic and elastic waves in inhomogeneous media. An initial boundary-value problem is replaced by a series of boundary-value problems for a constant…

Numerical Analysis · Mathematics 2011-01-25 Alexey G. Fatyanov , Andrew V. Terekhov

This paper proposes an efficient parallelised computation of field/circuit coupled systems co-simulated with the Waveform Relaxation (WR) technique. The main idea of the introduced approach lies in application of the parallel-in-time method…

Computational Physics · Physics 2020-03-17 Idoia Cortes Garcia , Iryna Kulchytska-Ruchka , Sebastian Schöps

In-silico investigation of skin permeation is an important but also computationally demanding problem. To resolve all scales involved in full detail will not only require exascale computing capacities but also suitable parallel algorithms.…

Computational Engineering, Finance, and Science · Computer Science 2015-10-19 Andreas Kreienbuehl , Arne Naegel , Daniel Ruprecht , Robert Speck , Gabriel Wittum , Rolf Krause

The Parareal algorithm allows to solve evolution problems exploiting parallelization in time. Its convergence and stability have been proved under the assumption of regular (smooth) inputs. We present and analyze here a new Parareal…

Numerical Analysis · Mathematics 2019-04-09 Martin J. Gander , Iryna Kulchytska-Ruchka , Innocent Niyonzima , Sebastian Schöps

A parareal algorithm based on an exponential $\theta$-scheme is proposed for the stochastic Schr\"odinger equation with weak damping and additive noise. It proceeds as a two-level temporal parallelizable integrator with the exponential…

Numerical Analysis · Mathematics 2018-03-28 Jialin Hong , Xu Wang , Liying Zhang

We present the design of a multiscale parareal method for kinetic equations in the fluid dynamic regime. The goal is to reduce the cost of a fully kinetic simulation using a parallel in time procedure. Using the multiscale property of…

Numerical Analysis · Mathematics 2025-02-06 Tino Laidin , Thomas Rey

We consider the parallel time integration of the linear advection equation with the Parareal and two-level multigrid-reduction-in-time (MGRIT) algorithms. Our aim is to develop a better understanding of the convergence behaviour of these…

Numerical Analysis · Mathematics 2019-10-15 Oliver A. Krzysik , Hans De Sterck , Scott P. MacLachlan , Stephanie Friedhoff

Applying parallel-in-time algorithms to multiscale Hamiltonian systems to obtain stable long time simulations is very challenging. In this paper, we present novel data-driven methods aimed at improving the standard parareal algorithm…

Numerical Analysis · Mathematics 2023-09-06 Rui Fang , Richard Tsai

Stochastic parareal (SParareal) is a probabilistic variant of the popular parallel-in-time algorithm known as parareal. Similarly to parareal, it combines fine- and coarse-grained solutions to an ordinary differential equation (ODE) using a…

Numerical Analysis · Mathematics 2023-03-13 Kamran Pentland , Massimiliano Tamborrino , T. J. Sullivan

In this paper the Micro-Macro Parareal algorithm was adapted to PDEs. The parallel-in-time approach requires two meshes of different spatial resolution in order to compute approximations in an iterative way to a predefined reference…

Numerical Analysis · Mathematics 2023-09-11 Benedict Philippi , Mahfuz Sarker Miraz , Thomas Slawig

In view of the existing limitations of sequential computing, parallelization has emerged as an alternative in order to improve the speedup of numerical simulations. In the framework of evolutionary problems, space-time parallel methods…

Numerical Analysis · Mathematics 2025-02-13 Andrés Arrarás , Francisco J. Gaspar , Iñigo Jimenez-Ciga , Laura Portero