  In this paper we present a class of high order accurate cell-centered
Arbitrary-Eulerian-Lagrangian (ALE) one-step ADER-WENO finite volume schemes
for the solution of nonlinear hyperbolic conservation laws on two-dimensional
unstructured triangular meshes. High order of accuracy in space is achieved by
a WENO reconstruction algorithm, while a local space-time Galerkin predictor
allows the schemes to be high order accurate also in time by using an
element-local weak formulation of the governing PDE on moving meshes. The mesh
motion can be computed by choosing among three different node solvers, which
are for the first time compared with each other in this article: the node
velocity may be obtained i) either as an arithmetic average among the states
surrounding the node, or, ii) as a solution of multiple one-dimensional
half-Riemann problems around a vertex, or, iii) by solving approximately a
multidimensional Riemann problem around each vertex of the mesh using the
genuinely multidimensional HLL Riemann. Once the vertex velocity and thus the
new node location has been determined by the node solver, the local mesh motion
is then constructed by straight edges connecting the vertex positions at the
old time level with the new ones at the next time level. If necessary, a
rezoning step can be introduced here to overcome mesh tangling or highly
deformed elements. We apply the high order algorithm presented in this paper to
the Euler equations of compressible gas dynamics as well as to the ideal
classical and relativistic MHD equations. We show numerical convergence results
up to fifth order of accuracy in space and time together with some classical
numerical test problems for each hyperbolic system under consideration.
