# Efficient Parallel-in-Time Solution of Time-Periodic Problems Using a Multi-Harmonic Coarse Grid Correction

## TL;DR

This paper introduces a parallel-in-time algorithm for nonlinear time-periodic problems that leverages multi-harmonic correction and a simplified Newton method to enable efficient, highly-parallelized solutions in both frequency and time domains.

## Contribution

It proposes a novel multi-harmonic coarse grid correction method that enhances parallelization in solving nonlinear time-periodic problems, improving efficiency over existing approaches.

## Key findings

- The method converges effectively for a 1D model problem.
- It outperforms existing methods in solving eddy current problems.
- Demonstrates scalability on a 3D transformer model.

## Abstract

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 computations via parallelization on the fine grid. The proposed approach reduces the complexity of the periodic Parareal solution by introducing a simplified Newton algorithm, which allows an additional parallelization on the coarse grid. In particular, at each Newton iteration a multi-harmonic correction is performed, which converts the block-cyclic periodic system in the time domain into a block-diagonal system in the frequency domain, thereby solving for each frequency component in parallel. The convergence analysis of the method is discussed for a one-dimensional model problem. The introduced algorithm and several existing solution approaches are compared via their application to the eddy current problem for both linear and nonlinear models of a coaxial cable. Performance of the considered methods is also illustrated for a three-dimensional transformer model.