  Due to the prevalence of magnetic fields in astrophysical environments,
magnetohydrodynamic (MHD) simulation has become a basic tool for studying
astrophysical fluid dynamics. To further advance the precision of MHD
simulations, we have developed a new simulation code that solves ideal
adiabatic or isothermal MHD equations with high-order accuracy. The code is
based on the finite-difference weighted essentially non-oscillatory (WENO)
scheme and the strong stability-preserving Runge-Kutta (SSPRK) method. Most of
all, the code implements a newly developed, high-order constrained transport
(CT) algorithm for the divergence-free constraint of magnetic fields,
completing its high-order competence. In this paper, we present the version in
Cartesian coordinates, which includes a fifth-order WENO and a fourth-order
five-stage SSPRK, along with extensive tests. With the new CT algorithm,
fifth-order accuracy is achieved in convergence tests involving the damping of
MHD waves in three-dimensional space. And substantially improved results are
obtained in magnetic loop advection and magnetic reconnection tests, indicating
a reduction in numerical diffusivity. In addition, the reliability and
robustness of the code, along with its high accuracy, are demonstrated through
several tests involving shocks and complex flows. Furthermore, tests of
turbulent flows reveal the advantages of high-order accuracy, and show the
adiabatic and isothermal codes have similar accuracy. With its high-order
accuracy, our new code would provide a valuable tool for studying a wide range
of astrophysical phenomena that involve MHD processes.
