# Nonlinear parallel-in-time multilevel Schur complement solvers for ordinary differential equations

## TL;DR

This paper introduces a parallel-in-time multilevel Schur complement solver for linear and nonlinear ODEs, enabling efficient, scalable solutions by leveraging multilevel time partitioning and innovative nonlinear strategies.

## Contribution

It presents a novel multilevel Schur complement approach for parallel-in-time ODE solving, including direct methods for linear problems and new strategies for nonlinear equations.

## Key findings

- The solver is weakly scalable with increasing computational resources.
- It efficiently handles both linear and nonlinear ODEs.
- Numerical experiments validate the method's effectiveness.

## Abstract

In this work, we propose a parallel-in-time solver for linear and nonlinear ordinary differential equations. The approach is based on an efficient multilevel solver of the Schur complement related to a multilevel time partition. For linear problems, the scheme leads to a fast direct method. Next, two different strategies for solving nonlinear ODEs are proposed. First, we consider a Newton method over the global nonlinear ODE, using the multilevel Schur complement solver at every nonlinear iteration. Second, we state the global nonlinear problem in terms of the nonlinear Schur complement (at an arbitrary level), and perform nonlinear iterations over it. Numerical experiments show that the proposed schemes are weakly scalable, i.e., we can efficiently exploit increasing computational resources to solve for more time steps the same problem.