Implementation of the CIP algorithm to magnetohydrodynamic simulations

An implementation of the Constrained Interpolation Profile (CIP) algorithm to magnetohydrodynamic (MHD) simulations is presented. First we transform the original momentum and magnetic induction equations to unfamiliar forms by introducing Elsässer variables [W.M. Elsässer, The hydromagnetic equations, Phys. Rev. (1950)]. In this formulation, while the compressional and pressure gradient terms remain as non-advective terms, the advective and magnetic stress terms are expressed in the form of an advection equation, which enables us to use the CIP algorithm. We have examined some 1D test problems using the code based on this formula. Linear Alfvén wave propagation tests reveal that the developed code is capable of solving any Alfvén wave propagation with only small numerical diffusion and phase errors up to k▵h=2.5k▵h=2.5 (where ▵h is the grid spacing). A shock tube test shows good agreement with a previous result with less numerical oscillation at the shock front and the contact discontinuity which are captured within a few grid points. Extension of the 1D code to the multi-dimensional case is straightforward. We have calculated the 3D nonlinear evolution of the Kelvin–Helmholtz instability (KHI) and compared the result with our previous study. We find that our new MHD code is capable of following the 3D turbulence excited by the KHI while retaining the solenoidal property of the magnetic field.