Local divergence-free polynomial interpolation on MAC grids

Abstract

Divergence-free vector fields play an important role in many types of problems, including the incompressible Navier-Stokes equations and the equations for magnetohydrodynamics. In the discrete setting, these fields are often obtained by projection, resulting in a discrete approximation of the continuous field that is discretely divergence-free. For many applications, such as tracing particles, this discrete field must then be extended to the entire region using interpolation. This interpolated field is continuous and differentiable (almost everywhere), but in general it will not be divergence-free. In this paper, we construct approximation schemes with the property that discretely divergence-free data interpolates to an analytically divergence-free vector field. Our focus is on data stored in a MAC grid layout that is divergence free under the second order central difference stencil, a case that is common in projection methods for the Navier-Stokes equations. While existing schemes with this property are known, they tend to be global (the interpolated value at a point depends on data stored on the grid far from that point) or discontinuous. We construct C0 and C1 continuous approximation schemes for 2D and 3D that are local and satisfy the divergence-free property. We also construct interpolating versions of the schemes that reproduce the MAC data at face centers. All eight schemes are explicit piecewise polynomials over small stencils.