A method-of-lines framework for energy stable arbitrary Lagrangian-Eulerian methods

TL;DR

This paper introduces a new method-of-lines framework for energy stable discretization of initial boundary value problems on moving domains, extending existing stationary domain methods to arbitrary mesh motion with guaranteed stability.

Contribution

It develops a semi-bounded operator framework for ALE methods, ensuring energy stability and automatic geometric conservation in moving domain simulations.

Findings

Energy estimates are equivalent in physical and reference domains.

The framework allows explicit or implicit time integration with stability guarantees.

Geometric conservation can be achieved automatically without affecting stability.

Abstract

We present a novel framework based on semi-bounded spatial operators for analyzing and discretizing initial boundary value problems on moving and deforming domains. This development extends an existing framework for well-posed problems and energy stable discretizations from stationary domains to the general case including arbitrary mesh motion. In particular, we show that an energy estimate derived in the physical coordinate system is equivalent to a semi-bounded property with respect to a stationary reference domain. The continuous analysis leading up to this result is based on a skew-symmetric splitting of the material time derivative, and thus relies on the property of integration-by-parts. Following this, a mimetic energy stable arbitrary Lagrangian-Eulerian framework for semi-discretization is formulated, based on approximating the material time derivative in a way consistent with discrete summation-by-parts. Thanks to the semi-bounded property, a method-of-lines approach using standard explicit or implicit time integration schemes can be applied to march the system forward in time. The same type of stability arguments applies as for the corresponding stationary domain problem, without regards to additional properties such as discrete geometric conservation. As an additional bonus we demonstrate that discrete geometric conservation, in the sense of exact free-stream preservation, can still be achieved in an automatic way with the new framework. However, we stress that this is not necessary for stability.