A new unsplit staggered mesh algorithm (USM) that solves multidimensional magnetohydrodynamics (MHD) on a staggered mesh is introduced and studied. Proper treatments of multidimensional flow problems are required for MHD simulations to avoid unphysical results that can even introduce numerical instability. The research work in this dissertation, which is based on an approach that combines the high-order Godunov method and the constrained transport (CT) scheme, uses such multidimensional consideration in a spatial reconstruction-evolution step.

The core problem of MHD simulation is the nonlinear evolution of solutions using well-designed algorithms that maintain the divergence-free constraint of the magnetic field components. The USM algorithm proposed in this dissertation ensures the solenoidal constraint by using Stokes' Theorem as applied to a set of induction equations. In CT-type of MHD schemes, one solves the discrete induction equations to proceed temporal evolutions of the staggered magnetic fields using electric fields. The accuracy of the computed electric fields therefore directly influence the solution quality of the magnetic fields. To meet this end, an accurate and improved electric field construction (IEC) scheme has been introduced as one of the essential parts of the current dissertation work.

Another important feature in this work is a development of a new algorithm that solves the induction equations with an added capability that controls numerical (anti)dissipations of the magnetic fields. This staggered dissipation-control differencing algorithm (SDDA) makes use of extra dissipation terms, for which their derivations are established from modified equations of the induction equations.

A series of comparison studies in a suite of numerical results of the USM-IEC-SDDA scheme will show a great deal of qualitative improvements in many stringent multidimensional MHD test problems.