Divergence-free Interpolation of Vector Fields From Point Values - Exact divB=0 in Numerical Simulations

TL;DR

This paper introduces a novel interpolation method for vector fields that ensures divergence-free conditions exactly, improving the accuracy of numerical simulations in astrophysics and electromagnetism.

Contribution

It presents a new class of finite-difference derivative operators that enforce divB=0 exactly through specific interpolation choices, clarifying divergence errors.

Findings

Demonstrates a divergence-free interpolation method for magnetic fields

Shows convergence of the proposed numerical implementation

Provides a framework applicable to any vector field

Abstract

In astrophysical magnetohydrodynamics (MHD) and electrodynamics simulations, numerically enforcing the divB=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 divB=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 divB=0 property. This principle clarifies the nature of divB != 0 errors. The principles and techniques demonstrated in this paper are particularly useful for the magnetic field, but can be applied to any vector field. This paper serves as a brief introduction to the method and demonstrates an implementation showing convergence.