A checkerboard-free, symmetry-preserving, conservative method for magnetohydrodynamics

Many industrial applications, such as nuclear fusion reactor design and aeronautics, require accurate numerical simulation tools, that conserve physical properties, for magnetohydrodynamic (MHD) flows at low magnetic Reynolds numbers. The interaction between the magnetic field and the conductive fluid often results in complex flow phenomena, which require conservative discretisation schemes. These schemes minimise numerical dissipation to improve accuracy, but can lead to the checkerboard problem if applied on collocated grid arrangements, which are necessary in complex geometries. An algorithm is constructed to simulate MHD flows, which relies on the symmetry-preserving discretisation to allow minimal numerical dissipation. Moreover, it quantifies the checkerboard problem, to dynamically balance it with numerical dissipation, only when necessary. Results for a laminar MHD Taylor–Green vortex case show the importance of the symmetry-preserving discretisation, for which the face-to-cell interpolation method is highlighted. Moreover, it is shown that the solver is virtually free from numerical dissipation on uniform meshes, and effectively counter-acts checkerboarding on skew meshes, where other solvers encounter difficulties. Turbulent MHD duct flow cases show the ability of the solver to generate accurate flow statistics and turbulent kinetic energy (TKE) budgets, while mitigating checkerboarding which lead to flow laminarisation in a comparative solver. These findings establish the presented framework as a reliable and efficient approach for MHD simulations at low magnetic Reynolds numbers, particularly where collocated grid arrangements and low numerical dissipation are required.