Modified-gradient methods for exact divergence-free in meshless magnetohydrodynamics

Abstract

We present a novel gradient regularization to completely eliminate the magnetic divergence error in meshless magnetohydrodynamics (MHD), which offers a high spatial resolution and conservative advantage, due to its Lagrangian nature. Comparing with the counterpart of constrained-gradient (CG) technique, we reform $\nabla \cdot \mathbf{B}=0$ by an implicit projection method to modify the magnetic-field gradients. The accuracy of modified-gradient (MG) method is verified and it achieves exact divergence-free results with round-off precision, by using tests of shock tube, 2D and 3D vortex, magneto-rotational instability, and especially, advection experiment, compared with CG method and the GIZMO code. It leads to noticeable improvement in pattern, amplitude and numerical dissipation of divergence error of magnetic field.

Figures (11)

• Figure 1: Brio-Wu shock tube. We present that density, velocity, magnetic field, internal energy, pressure and the map of the divergence error $\text{log}_{10}(h_i|\nabla\cdot \mathbf{B}_i|/|\mathbf{B}_i|)$ are evolving by GIZMO method (dark line), CG method (blue line), MG method (red line) and exact solution (dashed line) at $t=0.2$, respectively.
• Figure 2: Advection of a field loop. The first column is the initial state, other four columns show the results at $t = 20$, we compare four methods: MG, CG, GIZMO, POWELL as labelled (left-to-right). The ideal numerical result should maintain a perfect circle, say, scale and amplitude, keep the same as the initial state. Each column shows a map of magnetic pressure ${\mathbf B}_i^2/2*10^6$ (up), and the map of the divergence error $\text{log}_{10}(h_i|\nabla\cdot \mathbf{B}_i|/|\mathbf{B}_i|)$ (bottom), with values following the color bar. All test runs employ $256^2$ particles.
• Figure 3: 2D Orszag-Tang vortex. For each column, we compare four methods: MG, CT, CG, GIZMO, as labelled (top-to-bottom). Each three columns shows the density (left), magnetic pressure (middle) and the map of the divergence error $\text{log}_{10}(h_i|\nabla\cdot \mathbf{B}_i|/|\mathbf{B}_i|)$ (right), with values following the colorbar. All meshless runs contain $256^2$ particles and CT employs a $256^2$ grid. The results in left three columns at $t = 0.5$ and right three columns at $t = 4.0$.
• Figure 4: 2D Orszag-Tang vortex. We presents the conservation results of the MG method that total mass $m$ error, total momentum (two components $P1,P2$) error, total energy $E$ error, and the maximum magnetic flux $\max_i\{(V_i\nabla\cdot \mathbf{B}_i)\}$ for 2D Orszag-Tang vortex evolution over time [0,4].
• Figure 5: 2D Orszag-Tang vortex. We compare the MG method at three resolutions: MG128 ( $128^2$ particles), MG256 ( $256^2$ particles), and MG512 ( $512^2$ particles), from top to bottom. Each two column present density (left) and magnetic pressure (right), respectively. The results are shown at $t=0.5$ (left), $t=2.0$ (middle), and $t=4.0$ (right).