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 the vector field that is reconstructed in the abutting coarse mesh zone. If we retain just the four components that are present in the four fine mesh faces, then it means that each coarse mesh zone, where prolongation is to be carried out should be able to accommodate up to four pieces of information at each of its six faces. Using this information, one has to find three higher order volume-filling polynomials for the three vector field components in the coarse zone that is about to be refined. The polynomials should be such that they can match up to four pieces of information at the six faces of the coarse zone; they should do so while satisfying all the divergence-free constraints. This creates quite a mathematical puzzle, but at second order it was solved in Balsara [1]. (Subsequent work in Balsara et al. [11] has shown that the divergence-preserving constraints can also be accommodated.) Once the three polynomials are found, the vector field is analytically specified and it can be prolonged via simple areal integration and averaging to all the faces of any set of refined zones that replaces the coarse zone in question. Section 3.1 of Balsara [1] shows a very simple and easy to follow example of how this is done in two-dimensions and Sect. 4 of the same paper provides all the three-dimensional details.
The ideas developed can, of course, be applied to any vector field and any density. Setting the charge density to zero yields a divergence-free reconstruction strategy. While some parts of the fourth order formulation were presented in Balsara et al. [11], that formulation was not general enough to be applicable to the fourth order accurate divergence-preserving prolongation problem for AMR. The reason is evident from Fig. 2. To retain fourth order reconstruction of a facial component, one has to retain only ten moments within a face. (For example, in the z-face one would have to retain a constant term along with x- and y-moments, plus x2-, y2-, and xy-moments, and additionally the x3-, y3-, x2y-, and xy2-moments; for a total of ten moments.) However, consider a situation, where each fine mesh face had to retain not just the mean value of the normal component of the electric displacement but also the two linear moments in the two transverse directions, as shown in Fig. 2. That would mean that one has to have a minimum of twelve moments in each face. The formulation in Balsara et al. [11] does not have that extra flexibility, whereas the formulation presented here does have such flexibility.
The WENO-ADP reconstruction algorithm from the previous section can give us a high order representation of the vector field within a coarse zone. We first define a flux as an integration of the normal component of a vector field across an area. As a result, if a coarse zone is refined, the refinement procedure will generate new internal faces inside a coarse zone. To retain consistency on the coarse mesh, we do not touch the refined faces of a fine mesh that overlie a coarse mesh face. We only feel free to mildly touch-up the internal faces. In two-dimensions, Fig. 5 provides an example. Because the two-dimensional case is much easier to understand on our first encounter with the touch-up procedure, we present that first (by way of motivation) in Subsect. 4.1. In Subsect. 4.2, we present the three-dimensional case. (We are of course quick to add that the two-dimensional and three-dimensional cases differ substantially in their complexity, and only the three-dimensional case is truly valuable to applications scientists.) All through this section, we restrict attention to refinement ratios of two. In the next section, we will address the case, where the refinement ratio can exceed two.
We can see that the results described in Sect. 2 are analytically exact at fourth order. Furthermore, the mathematical construction works for divergence-free and divergence-preserving vector fields with refinement ratios of two. Therefore, in Subsect. 8.1, we demonstrate that divergence-free prolongation from a coarse mesh to a fine mesh with refinement ratio two that uses Sect. 2 indeed works. In Subsect. 8.2 we demonstrate that divergence-preserving prolongation from a coarse mesh to a fine mesh with refinement ratio two that uses Sect. 2 indeed works.
Subsections 8.1 and 8.2 show that a simple approach is available for the most basic need, which is to provide a higher order accurate divergence-free and divergence-preserving prolongation strategy that can work with refinement ratio of two. However, the results in those sections also provide us with a point of reference for the subsections that are to follow which show the versatility of the WENO-ADPT algorithm working at all orders and with different refinement ratios. We display refinement ratio of two, which exercises the results from Sect. 4, and refinement ratio of three, which exercises the results from Sect. 5. Subsections 8.3 and 8.4 show results from the WENO-ADPT algorithm for carrying out divergence-free and divergence-preserving prolongation, respectively, when refinement ratios of two are used. Subsections 8.5 and 8.6 show results from the WENO-ADPT algorithm for carrying out divergence-free and divergence-preserving prolongation, respectively, when refinement ratios of three are used. Subsection 8.7 shows the utility of using the finite volume WENO developed here for the prolongation of a scalar variable, or in fact, any vector field that does not have an associated constraint. 2b1af7f3a8