Divergent ~REPACK~ Free Vector Potential Pdfl

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In physics and mathematics, in the area of vector calculus, Helmholtz's theorem,[1][2] also known as the fundamental theorem of vector calculus,[3][4][5][6][7][8][9] states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into the sum of an irrotational (curl-free) vector field and a solenoidal (divergence-free) vector field; this is known as the Helmholtz decomposition or Helmholtz representation. It is named after Hermann von Helmholtz.[10]

A terminology often used in physics refers to the curl-free component of a vector field as the longitudinal component and the divergence-free component as the transverse component.[14] This terminology comes from the following construction: Compute the three-dimensional Fourier transform F ^ {\displaystyle {\hat {\mathbf {F} }}} of the vector field F {\displaystyle \mathbf {F} } . Then decompose this field, at each point k, into two components, one of which points longitudinally, i.e. parallel to k, the other of which points in the transverse direction, i.e. perpendicular to k. So far, we have

We examine hyperbolicity of general relativistic magnetohydrodynamics with divergence cleaning, a flux-balance law form of the model not covered by our earlier analysis. The calculations rely again on a dual-frame approach, which allows us to effectively exploit the structure present in the principal part. We find, in contrast to the standard flux-balance law form of the equations, that this formulation is strongly hyperbolic, and thus admits a well-posed initial value problem. Formulations involving the vector potential as an evolved quantity are then considered. Carefully reducing to first order, we find that such formulations can also be made strongly hyperbolic. Despite the unwieldy form of the characteristic variables we therefore conclude that of the free-evolution formulations of general relativistic magnetohydrodynamics presently used in numerical relativity, the divergence cleaning and vector potential formulations are preferred.

Adaptive mesh refinement (AMR) is the art of solving PDEs on a mesh hierarchy with increasing mesh refinement at each level of the hierarchy. Accurate treatment on AMR hierarchies requires accurate prolongation of the solution from a coarse mesh to a newly defined finer mesh. For scalar variables, suitably high-order finite volume WENO methods can carry out such a prolongation. However, classes of PDEs, such as computational electrodynamics (CED) and magnetohydrodynamics (MHD), require that vector fields preserve a divergence constraint. The primal variables in such schemes consist of normal components of the vector field that are collocated at the faces of the mesh. As a result, the reconstruction and prolongation strategies for divergence constraint-preserving vector fields are necessarily more intricate. In this paper we present a fourth-order divergence constraint-preserving prolongation strategy that is analytically exact. Extension to higher orders using analytically exact methods is very challenging. To overcome that challenge, a novel WENO-like reconstruction strategy is invented that matches the moments of the vector field in the faces, where the vector field components are collocated. This approach is almost divergence constraint-preserving, therefore, we call it WENO-ADP. To make it exactly divergence constraint-preserving, a touch-up procedure is developed that is based on a constrained least squares (CLSQ) method for restoring the divergence constraint up to machine accuracy. With the touch-up, it is called WENO-ADPT. It is shown that refinement ratios of two and higher can be accommodated. An item of broader interest in this work is that we have also been able to invent very efficient finite volume WENO methods, where the coefficients are very easily obtained and the multidimensional smoothness indicators can be expressed as perfect squares. We demonstrate that the divergence constraint-preserving strategy works at several high orders for divergence-free vector fields as well as vector fields, where the divergence of the vector field has to match a charge density and its higher moments. We also show that our methods overcome the late time instability that has been known to plague adaptive computations in CED.

The goal of this paper is to design methods for divergence-preserving prolongation of vector fields at high order. Furthermore, we are only interested in the three-dimensional case, because the two-dimensional problem is not of much interest in practical AMR applications. By default, unless it is specified, we will consider refinement ratios of two. However, the methods are general and, in the later sections, we will show that they can be used for refinement ratios that are larger than two. At second order, the problem was solved in Balsara [1] who presented a polynomial-based reconstruction strategy that could be used for prolongation. We present a very brief synopsis of that strategy so that the reader can appreciate the options available to us as we try to push towards higher order. Figure 2 shows a typical situation, where a fine mesh abuts a coarse mesh. If the coarse mesh has to be refined, we require that all the information about the four vector field components from the adjoining four fine mesh faces should be retained in