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We present a novel framework based on semi-bounded spatial operators foranalyzing and discretizing initial boundary value problems on moving anddeforming domains. This development extends an existing framework forwell-posed problems and energy stable discretizations from stationary domainsto the general case including arbitrary mesh motion. In particular, we showthat an energy estimate derived in the physical coordinate system is equivalentto a semi-bounded property with respect to a stationary reference domain. Thecontinuous analysis leading up to this result is based on a skew-symmetricsplitting of the material time derivative, and thus relies on the property ofintegration-by-parts. Following this, a mimetic energy stable arbitraryLagrangian-Eulerian framework for semi-discretization is formulated, based onapproximating the material time derivative in a way consistent with discretesummation-by-parts. Thanks to the semi-bounded property, a method-of-linesapproach using standard explicit or implicit time integration schemes can beapplied to march the system forward in time. The same type of stabilityarguments applies as for the corresponding stationary domain problem, withoutregards to additional properties such as discrete geometric conservation. As anadditional bonus we demonstrate that discrete geometric conservation, in thesense of exact free-stream preservation, can still be achieved in an automaticway with the new framework. However, we stress that this is not necessary forstability.

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