Stability analysis of the free-surface Stokes problem and an unconditionally stable explicit scheme

TL;DR

This paper provides a theoretical stability analysis of the coupled free-surface Stokes problem for viscous flows, introduces a stabilization method for explicit schemes ensuring unconditional stability and volume conservation, and validates findings through numerical experiments.

Contribution

It offers a novel stabilization term for explicit Euler methods that guarantees unconditional stability and volume conservation in free-surface Stokes flow simulations.

Findings

Theoretical proof of stability and conservation properties.

Development of a stabilization term for explicit Euler schemes.

Numerical experiments confirming theoretical predictions.

Abstract

Accurate simulations of ice sheet dynamics, mantle convection, lava flow, and other highly viscous free-surface flows involve solving the coupled Stokes/free-surface equations. In this paper, we theoretically analyze the stability and conservation properties of the weak form of this system for Newtonian fluids and non-Newtonian fluids, at both the continuous and discrete levels. We perform the fully discrete stability analysis for finite element methods used in space with explicit and implicit Euler time-stepping methods used in time. Motivated by the theory, we propose a stabilization term designed for the explicit Euler discretization, which ensures unconditional time stability and permits conservation of the domain volume. Numerical experiments validate and support our theoretical findings.