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) an 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.3

BOUT++: A framework for parallel plasma fluid simulations

“…These can be compared with figure 11 in [48], figure 3 in [49], and figure 9 in [50]. Figure 9 shows that whilst the BOUT++ solutions are very close to those from specialised MHD codes, it is susceptible to oscillations at the top of shocks.…”

An improved CE/SE scheme for incompressible multiphase flows and its applications

“…Typically, both independent flow variables and their derivatives are treated as unknowns and are solved for simultaneously. The CE/SE method has been used for accurate numerical solutions involving shocks (Chang, 1995), viscous flows (Guo et al, 2004), magneto-hydro-dynamic (MHD) flows (Zhang et al, 2006), multi-material elastic-plastic flows (Wang et al, 2009) and dilute gas-particle flows (Wang et al, 2011), etc. Recently, Zhang et al (2002 proposed the CE/SE scheme by using quadrilateral and hexahedral meshes.…”

A local space–time conservation scheme and its application in shock wave propagation

“…The CE/SE method was originally proposed by Chang and co-workers [7][8][9][10][11][12][13][14] which was a completely new numerical framework for solving hyperbolic conservation equations. This new approach differs substantially in both concept and methodology from the well-established methods.…”

“…Theoretical analysis and numerical results demonstrate that the CE/SE method has quite high accuracy and resolution of strong discontinuity [9]. To date, the CE/SE method has achieved great success in simulations of sound wave propagating [11], aero-acoustics [9,15], steady viscous flow [16], complex supersonic flow [17], chemical reactive flow [14,[18][19][20], magneto-hydrodynamics [12], multimaterial elastic-plastic flows [21], spall fracture [22] and so on.…”