Piecewise parabolic method on a local stencil for ideal magnetohydrodynamics

A numerical scheme based on the piecewise parabolic method on a local stencil (PPML) is proposed for solving the ideal magnetohydrodynamic (MHD) equations. The method makes use of the conservation of Riemann invariants along the characteristics of the MHD equations. As a result, a local stencil can be used to construct a numerical solution. This approach improves the dissipative properties of the numerical scheme and is convenient in the case of adaptive grids. The basic stages in the design of the scheme are illustrated in the two-dimensional case. The conservation of the solenoidal property of the magnetic field is discussed. The scheme is tested using several typical MHD problems.