Solving the MHD equations by the space–time conservation element and solution element method

Abstract

We apply the space–time conservation element and solution element (CESE) method to solve the ideal MHD equations with special emphasis on satisfying the divergence free constraint of magnetic field, i.e., ∇ · B = 0. In the setting of the CESE method, four approaches are employed: (i) the original CESE method without any additional treatment, (ii) a simple corrector procedure to update the spatial derivatives of magnetic field B after each time marching step to enforce ∇ · B = 0 at all mesh nodes, (iii) a constraint-transport method by using a special staggered mesh to calculate magnetic field B, and (iv) the projection method by solving a Poisson solver after each time marching step. To demonstrate the capabilities of these methods, two benchmark MHD flows are calculated: (i) a rotated one-dimensional MHD shock tube problem and (ii) a MHD vortex problem. The results show no differences between different approaches and all results compare favorably with previously reported data.

Introduction

While many computational fluid dynamics (CFD) methods have been successfully developed for gas dynamics, extension of these methods for solving the Magneto-Hydro-Dynamic (MHD) equations involves unique requirements and poses greater challenges [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14]. In particular, for multi-dimensional MHD problems, it is critical to maintain the divergence-free constraint of magnetic field, i.e., ∇ · B = 0, at all locations in the space–time domain. Analytically, the constraint is ensured if it is satisfied in the initial condition. However, it has been difficult to maintain this constraint in calculating evolving MHD problems. Violating the constraint allows numerical errors to be accumulated over the computational time, leading to erroneous solutions and/or numerical instability.

To satisfy ∇ · B = 0, a special treatment directly incorporated into the CFD method employed is often required. Special treatments have been categorized into three groups: (i) The projection method, e.g., Brackbill and Barnes [5]: At each time step, the method solves a Poisson equation to update the magnetic field to enforce ∇ · B = 0. (ii) The eight-wave formulation by Powell [6]: ∇ · B is not treated as zero in deriving the MHD equations, leading to additional source/sink terms in equations for B. The CFD solver employed would activate the sink/source terms to counter the unbalanced ∇ · B in numerical solutions. (iii) The constrained-transport procedure, e.g., Evans and Hawley [7], Dai and Woodward [8], Balsara and Spice [9], and Tóth [12], based on the use of staggered mesh to enforce the constraint at certain spots of the control volume. Various versions of these three approaches have been developed to solve the MHD equations in multiple spatial dimensions [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]. Recently, these methods have been assessed and summarized by Tóth [12].

In the present paper, we report the application of the space–time conservation element and solution element (CESE) method [15], [16], [17], [18], [19], [20] to solve the two-dimensional MHD equations. Four approaches are employed: (i) the original CESE method without any additional treatment for ∇ · B = 0, (ii) a simple modification procedure to update the spatial derivatives of B after each time marching step such that ∇ · B = 0 is enforced at all mesh nodes, (iii) an extended CESE method based on the constraint-transport procedure, and (iv) the projection method coupled with the CESE method. The approach (i) is trivial. Nevertheless, its results are comparable with other results by the three other approaches. Approaches (ii) and (iii) are new schemes for ∇ · B = 0. Approach (iv), the projection method, is a conventional and reliable approach to impose ∇ · B = 0. All results in the present paper compare well with previously published data.

The rest of the paper is arranged as follows. Section 2 illustrates the governing equations. Section 3 provides a brief review of the CESE method for two-spatial-dimensional problems. Section 4 shows the new CESE schemes, i.e., approaches (ii) and (iii), for ∇ · B = 0. Section 5 provides the results and discussions. We then offer conclusions and provide cited references.