Abstract We present a new magnetohydrodynamic (MHD) simulation package with the aim of providing accurate numerical solutions to astrophysical phenomena where discontinuities, shock waves, and turbulence are inherently important. The code implements the Harten–Lax–van Leer–discontinuitues (HLLD) approximate Riemann solver, the fifth-order-monotonicity-preserving interpolation (MP5) scheme, and the hyperbolic divergence cleaning method for a magnetic field. This choice of schemes has significantly improved numerical accuracy and stability, and saved computational costs in multidimensional problems. Numerical tests of one- and two-dimensional problems show the advantages of using the high-order scheme by comparing with results from a standard second-order total variation diminishing monotonic upwind scheme for conservation laws (MUSCL) scheme. The present code enables us to explore the long-term evolution of a three-dimensional accretion disk around a black hole, in which compressible MHD turbulence causes continuous mass accretion via nonlinear growth of the magneto-rotational instability (MRI). Numerical tests with various computational cell sizes exhibits a convergent picture of the early nonlinear growth of the MRI in a global model, and indicates that the MP5 scheme has more than twice the resolution of the MUSCL scheme in practical applications.