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In this paper, we extend the unsplit staggered mesh scheme (USM) for 2Dmagnetohydrodynamics (MHD) (Lee and Deane, 2009) to a full 3D MHD scheme. Thescheme is a finite-volume Godunov method consisting of a constrained transport(CT) method and an efficient and accurate single-step, directionally unsplitmultidimensional data reconstruction-evolution algorithm, which extends theoriginal 2D corner transport upwind (CTU) method (Colella, 1990). We presenttwo types of data reconstruction-evolution algorithms for 3D: (1) a reduced CTUscheme and (2) a full CTU scheme. The reduced 3D CTU scheme is a variant of asimple 3D extension of the 2D CTU method by Colella (1990) and is considered asa direct extension from the 2D USM scheme. The full 3D CTU scheme is ourprimary 3D solver which includes all multidimensional cross-derivative termsfor stability. The latter method is logically analogous to the 3D unsplit CTUmethod by Saltzman. The major novelties in our algorithms are twofold. First,we extend the reduced CTU scheme to the full CTU scheme which is able to runwith CFL numbers close to unity. Both methods utilize the transverse updatetechnique developed in the 2D USM algorithm to account for transverse fluxeswithout solving intermediate Riemann problems, which in turn givescost-effective 3D methods by reducing the total number of Riemann solves. Theproposed algorithms are simple and efficient especially when includingmultidimensional MHD terms that maintain in-plane magnetic field dynamics.Second, we introduce a new CT scheme that makes use of proper upwindinformation in taking averages of electric fields.

A Solution Accurate, Efficient and Stable Unsplit Staggered Mesh Scheme for Three Dimensional Magnetohydrodynamics