  Magneto-hydrodynamics is one of the foremost models in plasma physics with
applications in inertial confinement fusion, astrophysics and elsewhere.
Advanced numerical methods are needed to get an insight into the complex
physical phenomena. The classical Lagrangian methods are typically limited to
the low orders of convergence and suffer from violation of the divergence-free
condition for magnetic field or conservation of the invariants. This paper is
the first part of a new series about high-order non-ideal
magneto-hydrodynamics, where a multi-dimensional conservative Lagrangian method
based on curvilinear finite elements is presented. The condition on zero
divergence of magnetic field and conservation of mass, momentum, magnetic flux
and the total energy are satisfied exactly. The curvilinear elements prevent
entangling of the computational mesh and its imprinting into the solution. A
high-order conservative time integration is applied, where an arbitrary order
of convergence is attained for problems of ideal magneto-hydrodynamics. The
resistive magnetic field diffusion is solved by an implicit scheme. Description
of the method is given and multiple test problems demonstrating properties of
the scheme are performed. The construction of the method and possible future
directions of development are discussed.
