# Convergence analysis for parallel-in-time solution of hyperbolic systems

## TL;DR

This paper analyzes the convergence of multigrid methods for parallel-in-time solutions of hyperbolic PDEs, introducing a combined analysis approach that improves understanding and potential stability of these methods.

## Contribution

It develops a combined analysis framework integrating reduction analysis and semi-algebraic mode analysis for hyperbolic problems, enhancing convergence insights.

## Key findings

- Combined analysis improves stability predictions for hyperbolic problems.
- Numerical validation confirms the effectiveness of the combined analysis.
- Insights apply to linear advection and elasticity models.

## Abstract

Parallel-in-time algorithms have been successfully employed for reducing time-to-solution of a variety of partial differential equations, especially for diffusive (parabolic-type) equations. A major failing of parallel-in-time approaches to date, however, is that most methods show instabilities or poor convergence for hyperbolic problems. This paper focuses on the analysis of the convergence behavior of multigrid methods for the parallel-in-time solution of hyperbolic problems. Three analysis tools are considered that differ, in particular, in the treatment of the time dimension: (1) space-time local Fourier analysis, using a Fourier ansatz in space and time, (2) semi-algebraic mode analysis, coupling standard local Fourier analysis approaches in space with algebraic computation in time, and (3) a two-level reduction analysis, considering error propagation only on the coarse time grid. In this paper, we show how insights from reduction analysis can be used to improve feasibility of the semi-algebraic mode analysis, resulting in a tool that offers the best features of both analysis techniques. Following validating numerical results, we investigate what insights the combined analysis framework can offer for two model hyperbolic problems, the linear advection equation in one space dimension and linear elasticity in two space dimensions.