What is it about?

A conservative finite difference method was constructed with consideration of the induced magnetic field for incompressible magnetohydrodynamics flows. This study compared the difference in energy conservation properties among the three methods of calculating the Lorentz force. The Lorentz forces are calculated in conservative and non-conservative forms, and both compact and wide-range interpolations of magnetic flux density are used to calculate the non-conservative Lorentz force. The compact interpolation method proposed in this study can convert between conservative and non-conservative forms of the Lorentz force even when using the finite difference method. The present numerical method improves the conservation of the transport quantity. This study analyzed five models and verified the accuracy and convergence of the present numerical method. From the viewpoint of conservation of total energy in an ideal inviscid periodic MHD flow, we consider that the calculation using compact interpolation for the Lorentz force is appropriate. This method preserves the total energy even on non-uniform grids. Moreover, the divergence-free condition of the magnetic flux density is discretely satisfied even without the correction of the magnetic flux density. The present numerical method can capture the Hartmann layer in the propagation of the Alfvén wave and accurately capture the tendency of energy attenuation in the analysis of the Taylor decaying vortex under magnetic fields. Analysis of the Orszag-Tang vortex reveals energy dissipation processes and the generation of high current densities.

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Why is it important?

In the analysis of magnetohydrodynamic flow, the Lorentz force significantly affects energy properties because the work generated by the Lorentz force changes the kinetic and magnetic energies. Therefore, the Lorentz force and energy conversion should be predicted accurately. Several energy conservation schemes have been proposed and validated. However, the influences of the Lorentz force discretization on the conservation and conversion of energy have not been clarified. This study constructed an energy-conserving finite difference scheme and investigated the influence of a discretization method of the Lorentz force on energy conservation. The present numerical method can discretely transform the Lorentz force between conservative and non-conservative forms. Moreover, it is clarified that energy is conserved in inviscid magnetohydrodynamic flows.


Numerical analysis of magnetohydrodynamic flow is a very difficult problem because energy conversion occurs. Various numerical methods have been proposed for the analysis of magnetohydrodynamic flow. However, the influence of the Lorentz force discretization method on energy conservation has not been clarified. In this study, we focused on the influence of the Lorentz force calculation method on energy conservation and constructed an energy-conserving finite difference scheme. The present numerical method has excellent energy conservation properties and can accurately predict energy conversion. Therefore, this method can contribute to understanding complex unsteady magnetohydrodynamic flows.

Professor Hideki Yanaoka
Iwate Daigaku

Read the Original

This page is a summary of: Influences of conservative and non-conservative Lorentz forces on energy conservation properties for incompressible magnetohydrodynamic flows, Journal of Computational Physics, July 2023, Elsevier,
DOI: 10.1016/j.jcp.2023.112372.
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