A time adaptive multirate Quasi-Newton waveform iteration for coupled problems

TL;DR

This paper introduces a novel time adaptive multirate Quasi-Newton waveform iteration method for coupled PDE problems, enhancing convergence and efficiency in partitioned simulations with adaptive time stepping.

Contribution

It extends Quasi-Newton waveform iteration to adaptive time settings, providing a new approach for efficient coupled problem simulations.

Findings

Demonstrates improved convergence over existing methods.

Shows efficiency gains in heat transfer and fluid-structure interaction cases.

Validates the method's effectiveness through numerical experiments.

Abstract

We consider waveform iterations for dynamical coupled problems, or more specifically, PDEs that interact through a lower dimensional interface. We want to allow for the reuse of existing codes for the subproblems, called a partitioned approach. To improve computational efficiency, different and adaptive time steps in the subsolvers are advisable. Using so called waveform iterations in combination with relaxation, this has been achieved for heat transfer problems earlier. Alternatively, one can use a black box method like Quasi-Newton to improve the convergence behaviour. These methods have recently been combined with waveform iterations for fixed time steps. Here, we suggest an extension of the Quasi-Newton method to the time adaptive setting and analyze its properties. We compare the proposed Quasi-Newton method with state of the art solvers on a heat transfer test case, and a complex mechanical Fluid-Structure interaction case, demonstrating the methods efficiency.