  A high-fidelity finite volume scheme based on the BVD (boundary variation
diminishing) concept is proposed in this study to solve the ideal
magnetohydrodynamics (MHD) equations. A hybrid spatial reconstruction profile,
consisting of a quadratic polynomial and a steepness-adjustable hyperbolic
tangent function, is adopted to reproduce the accurate solutions of the complex
magnetohydrodynamics flows. The BVD principle is used to find a optimal
combination of these two types of spatial reconstructions by comparing the
variations of the interface values interpolated in two adjacent cells, aiming
to remove the non-physical oscillations around discontinuities by switching the
quadratic polynomial to a step-shaped function. Additionally, a constrained
transport (CT) method is applied in this study to assure the non-divergent
solution of the magnetic field. The widely used numerical tests in one and two
dimensional cases were checked in this study. The numerical results can
retrieve the accuracy of a $3^{rd}$-order linear scheme in the convergence test
for both advection and MHD equations and capture the strong shock waves in MHD
flows without spurious oscillations. In comparison with the results of a
$3^{rd}$-order WENO (weighted essentially non-oscillatory) scheme, the proposed
scheme gains more accurate solutions not only for the strong discontinuities
but also the smooth structures across scales. To our knowledge, this is the
first attempt to build a high-fidelity model for ideal MHD equations by a BVD
algorithm. Numerical results are competitive to those of existing advanced
models and thus the BVD algorithm has promising potentials to build practical
models for various MHD flows.
