Divergence-free interpolation of vector fields from point values — exact ∇ ⋅B = 0 in numerical simulations

In astrophysical magnetohydrodynamics (MHD) and electrodynamics simulations, numerically enforcing the ∇·B= 0 constraint on the magnetic field has been difficult. We observe that for point-based discretization, as used in finite-difference type and pseudo-spectral methods, the ∇·B= 0 constraint can be satisfied entirely by a choice of interpolation used to define the derivatives of B. As an example we demonstrate a new class of finite-difference-type derivative operators on a regular grid which has the ∇·B= 0 property. This principle clarifies the nature of ∇·B≠ 0 errors. The principles and techniques demonstrated in this Letter are particularly useful for the magnetic field, but can be applied to any vector field. This Letter serves as a brief introduction to the method and demonstrates an implementation showing convergence.