A priori error analysis of multirate time-stepping schemes for two-phase flow problems

TL;DR

This paper provides a priori error estimates for multirate time-stepping schemes applied to coupled multiphysics problems like two-phase flows, demonstrating optimal error bounds and decoupling advantages.

Contribution

It introduces a novel a priori error analysis for multirate schemes in coupled problems, ensuring optimal error estimates and decoupling of subproblems.

Findings

Optimal error estimates for velocity in coupled problems.

Suboptimal error estimates for pressure.

Decoupling of subproblems enables adaptive lattice selection.

Abstract

We present a priori error estimates for a multirate time-stepping scheme for coupled differential equations. The discretization is based on Galerkin methods in time using two different time meshes for two parts of the problem. We aim at surface coupled multiphysics problems like two-phase flows. Special focus is on the handling of the interface coupling to guarantee a coercive formulation as key to optimal order error estimates. In a sequence of increasing complexity, we begin with the coupling of two ordinary differential equations, coupled heat conduction equation, and finally a coupled Stokes problem. For this we show optimal multi-rate estimates in velocity and a suboptimal result in pressure. The a priori estimates prove that the multirate method decouples the two subproblems exactly. This is the basis for adaptive methods which can choose optimal lattices for the respective subproblems.