  In this article a new high order accurate cell-centered
Arbitrary-Lagrangian-Eulerian (ALE) Godunov-type finite volume method with
time-accurate local time stepping (LTS) is presented. The method is by
construction locally and globally conservative. The scheme is based on a
one-step predictor-corrector methodology in space-time and uses three main
building blocks: First, a high order piecewise polynomial WENO reconstruction,
to obtain a high order data representation in space from the known cell
averages of the underlying finite volume scheme. Second, a high order
space-time Galerkin predictor step based on a weak formulation of the governing
PDE on moving control volumes. Third, a high order one-step finite volume
scheme, based directly on the integral formulation of the conservation law in
space-time. The algorithm being entirely based on space-time control volumes
naturally allows for hanging nodes also in time, hence in this framework the
implementation of a consistent and conservative time-accurate LTS becomes very
natural and simple. The method is validated on some classical shock tube
problems for the Euler equations of compressible gasdynamics and the
magnetohydrodynamics equations (MHD). The performance of the new scheme is
compared with a classical high order ALE finite volume scheme based on global
time stepping. To the knowledge of the author, this is the first high order
accurate Lagrangian finite volume method ever presented together with a
conservative and time-accurate local time stepping feature.
