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Due to the prevalence of magnetic fields in astrophysical environments,magnetohydrodynamic (MHD) simulation has become a basic tool for studyingastrophysical fluid dynamics. To further advance the precision of MHDsimulations, we have developed a new simulation code that solves idealadiabatic or isothermal MHD equations with high-order accuracy. The code isbased on the finite-difference weighted essentially non-oscillatory (WENO)scheme and the strong stability-preserving Runge-Kutta (SSPRK) method. Most ofall, 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 inCartesian coordinates, which includes a fifth-order WENO and a fourth-orderfive-stage SSPRK, along with extensive tests. With the new CT algorithm,fifth-order accuracy is achieved in convergence tests involving the damping ofMHD waves in three-dimensional space. And substantially improved results areobtained in magnetic loop advection and magnetic reconnection tests, indicatinga reduction in numerical diffusivity. In addition, the reliability androbustness of the code, along with its high accuracy, are demonstrated throughseveral tests involving shocks and complex flows. Furthermore, tests ofturbulent flows reveal the advantages of high-order accuracy, and show theadiabatic and isothermal codes have similar accuracy. With its high-orderaccuracy, our new code would provide a valuable tool for studying a wide rangeof astrophysical phenomena that involve MHD processes.

HOW-MHD: A High-Order WENO-Based Magnetohydrodynamic Code with a High-Order Constrained Transport Algorithm for Astrophysical Applications