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Charge-conserving hybrid methods for the Yang-Mills equations

03/23/2020
by   Yakov Berchenko-Kogan, et al.
0

The Yang-Mills equations generalize Maxwell's equations to nonabelian gauge groups, and a quantity analogous to charge is locally conserved by the nonlinear time evolution. Christiansen and Winther observed that, in the nonabelian case, the Galerkin method with Lie algebra-valued finite element differential forms appears to conserve charge globally but not locally, not even in a weak sense. We introduce a new hybridization of this method, give an alternative expression for the numerical charge in terms of the hybrid variables, and show that a local, per-element charge conservation law automatically holds.

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1. Introduction

Maxwell’s equations satisfy a charge conservation law: assuming no current, the charge density is constant in time. The equation is often viewed as a constraint, but since it is automatically preserved by the evolution of , the constraint need not be “enforced” in any way. The Yang–Mills equations can be seen as a nonlinear, nonabelian generalization of Maxwell’s equations, and an analogous charge conservation law holds in this more general context. One would like this conservation law to continue to hold in numerical simulations of the equations, but this is not necessarily the case, even for Maxwell’s equations. Christiansen and Winther address the issue of constraint preservation in [8], where they write that “The Yang–Mills equations appear relatively ripe for numerical analysis and could therefore serve as a stepping stone toward the successful simulation of more complicated equations,” such as Einstein’s equations of general relativity, whose nonlinear evolution also preserves physically important constraints.

In their paper, Christiansen and Winther observe that a standard Galerkin semidiscretization of the Yang–Mills equations only yields conservation of the total charge on the whole domain. Locally, charge is not conserved, as they illustrate in Figure 3 of their paper. Christiansen and Winther solve this problem with a constrained scheme that artificially imposes the charge conservation constraint. A different low-order charge-conserving method, based on lattice gauge theory, was given by Christiansen and Halvorsen [7]; this method preserves the constraint automatically but requires commiting a “variational crime” by modifying the Yang–Mills variational principle.

In contrast, we present an alternate approach, which automatically preserves a local charge conservation law without modifying the Yang–Mills variational principle. As in our work on Maxwell’s equations in [4]

, we consider the domain-decomposed problem, where we use discontinuous finite element spaces for our vector and scalar potentials, and then impose inter-element continuity and boundary conditions with Lagrange multipliers

and . Using the hybrid variable , we obtain an expression for the charge. While we are not able to get strong charge conservation when we semidiscretize, as we did for Maxwell’s equations, we are able to get a local conservation law: the total charge on each element is conserved.

The paper is structured as follows. In Section 2, we introduce our notation and discuss the Yang–Mills equations, leading up to the conservation of total charge in the Galerkin semidiscretization observed by Christiansen and Winther. In Section 3, we describe our domain-decomposed numerical scheme for the Yang–Mills equations and prove that it satisfies a local charge conservation property. Finally, in Section 4, we discuss our numerical implementation and illustrate with examples.

2. Preliminaries

2.1. Lie algebra-valued differential forms

In this section, we introduce Lie algebra-valued differential forms, largely following [9].

Let be a compact Lie group with Lie algebra . Let denote the Lie bracket on . Such a Lie algebra always has an invariant inner product with the property that for all .

Any compact Lie group can be represented as a group of unitary matrices, whose algebra consists of skew-Hermitian matrices with the commutator bracket

. For simplicity of notation, we will thus view both and as sets of matrices, in which case we can choose the inner product to simply be . where denotes the conjugate transpose of .

Definition .

Let be a bounded Lipschitz domain. A -valued -form on is a section of the bundle . We will denote the space of -valued -forms by . We will denote the Lebesgue spaces of sections of by .

Example .

In the setting of electromagnetism, , the unit complex numbers. Then , the purely imaginary numbers. Thus, in this setting, a -valued -form is simply an ordinary -form times the imaginary unit . The Lie bracket is identically zero, and the inner product is simply .

The space is spanned by forms , where is a real-valued -form and is a constant element of . With this decomposition, we can define several operations on -valued -forms.

Definition .

Given and , define

and extend these operations to arbitrary -valued forms by linearity.

In the case where either or is a -form, i.e., just a Lie algebra-valued function, we will often write and instead of and .

We have the following identities for -valued forms.

Proposition .

For , , we have the Leibniz rules

(1)
(2)

and the commutativity relations

(3)
(4)

Additionally, given ,

(5)
(6)
Proof.

It suffices to prove these identities for forms of the type , , , since they extend to arbitrary forms by linearity.

The Leibniz rules (1) and (2) follow immediately from the Leibniz rule for ordinary real-valued forms.

The commutativity relations (3) and (4) follow from , together with the antisymmetry of and symmetry of , respectively.

Finally, (5) and (6) follow from , together with the Jacobi identity for and the invariance property of , respectively. ∎

In the classical formulation of electromagnetics, the electric field and electric flux density are vector fields, where is the

electric permittivity tensor

. Likewise, the magnetic flux density and magnetic field are vector fields, where is the magnetic permeability tensor. When expressed in terms of differential forms, and are -forms, and are -forms, and and correspond to Hodge operators mapping -forms and -forms to -forms and -forms, respectively. In vacuum, with appropriately chosen units, each of these is simply the ordinary Hodge star operator . For more on the differential forms point of view for finite element methods in computational electromagnetics, see Hiptmair [10] and references therein.

This motivates the following generalized notion of electric permittivity and magnetic permeability, in arbitrary dimension , for both ordinary and -valued differential forms.

Definition .

The electric permittivity tensor and magnetic permeability tensor are pointwise symmetric isomorphisms

for each . The symmetry of and is in the sense that

We can extend these isomorphisms to maps

by ignoring the Lie algebra coefficient; that is .

As before, these operators have (anti)symmetry properties.

Proposition .

In particular, for and for .

Proof.

As before, we can prove these claims for basic tensors and using the symmetry of , , and and the antisymmetry of . We then extend to general and by linearity. ∎

2.2. Connections, curvature, and the exterior covariant derivative

We now discuss connections, again following [9]. As in [8], we restrict our attention to the trivial bundle case. In this setting, a connection is just a -valued one-form.

Definition .

Let . The curvature of , denoted , is defined by

The exterior covariant derivative with respect to , denoted , is defined by

Example .

In the setting of electromagnetism with , the Lie algebra has trivial commutator . Thus, and .

Unlike in electromagnetism, . Instead, , in the following sense:

Proposition .

Let . Then

Additionally, we will make use of the Bianchi identity

Proposition (Bianchi identity).

We have a product rule for the exterior covariant derivative

Proposition .

If , and , then

Proof.

The Leibniz rule (1) gives , while (5) implies . Adding these together gives the claimed identity. ∎

Finally, we can integrate by parts using the exterior covariant derivative.

Proposition .

Let and . Then

Proof.

The first line follows from Stokes’s theorem and the Leibniz rule (2). The second line follows from the fact that , which is a special case of (6). ∎

2.3. Electric and magnetic fields

In order to define the Yang–Mills analogues of the scalar and vector potentials and the electric and magnetic fields, we will need some regularity assumptions. We define the following spaces

Definition .

Let

We let and denote those forms and , respectively, whose tangential traces vanish on the boundary of .

The regularity assumptions on ensure that . The regularity assumptions on ensure that for , which will be necessary later to show charge conservation. See Equation (13) and Section 3.3.

We can now define the Yang–Mills analogues of the scalar and vector potentials, the electric field, and the magnetic flux density. Note that we still refer to these as “scalar” and “vector” potentials, even though they are actually -valued forms in this generalized setting. Here and henceforth, we employ the commonly-used “dot” notation for partial differentiation with respect to time, e.g., means .

Definition .

Let the scalar potential be a curve in and let the vector potential be a curve in . Then define the electric field and magnetic flux density by

From this, we immediately see that and .

Example .

Recall that in the setting of electromagnetism with , a -valued one-form is a real-valued one-form times the imaginary unit . By omitting the imaginary unit and converting the one-form to a vector field, we obtain a correspondence between the vector potential expressed as a -valued one-form and the vector potential expressed clasically as a vector field. Similarly, the scalar potential in this notation is a function with purely imaginary values. By omitting the imaginary unit, we obtain the usual real-valued scalar potential.

Recall that when , we have and , so the equations for and simplify to and . Converting these differential forms to vector fields, we obtain the usual equations and .

Using the identities and , we obtain that

In the setting of electromagnetism, these equations correspond to the Maxwell equations and .

To define the electric flux density and the magnetic field , we utilize the electric permittivity tensor and magnetic permeability tensor of Section 2.1. We assume that both and are maps.

Definition .

Let

From these definitions, and need only be curves in . We make the stronger assumption that is in fact a curve in .

2.4. The Yang–Mills Lagrangian

For this discussion, we will set the current to be zero, and we will view the charge density as a curve in .

Definition .

The Yang–Mills Lagrangian is

(7)

Note that each term is a real-valued -form in at least the Lebesgue space, so we can indeed integrate this expression over .

The Euler–Lagrange equations are

(8a)
(8b)

These are weak expressions of the Yang–Mills equations

(9a)
(9b)
Example .

In the setting of electromagnetism with , recall that and that . Thus, the Yang–Mills equations in this context are

which are differential form expressions of Maxwell’s equations,

The Yang–Mills equations imply a charge conservation law.

Proposition .

Equations (9) imply that satisfies

In particular is conserved.

Proof.

We compute

Then,

2.5. Gauge symmetry

Definition .

A gauge transformation is a time-dependent -valued field on . That is, a gauge transformation is a function . A gauge transformation acts on the vector and scalar potentials by the transformation

To explain the notation, recall that we view and as subsets of matrices, so means , where the expression is matrix multiplication. Meanwhile, fixing a point in time and viewing as a map , we take the derivative to obtain a map . Thus we can view as a -valued one-form, and so is a one-form with values in . Similarly, fixing a point in space, we can view as map . The velocity of this path is a tangent vector , and, again, is in .

Example .

In the setting of electromagnetism with , recall that a -valued -form is simply a real-valued -form times the imaginary unit . Let be a scalar field on . Then, setting , we see that is a gauge transformation, and

matching the formula for gauge transformations in electromagnetism. Seeing as a vector field and as a scalar field, this is , leaving and invariant.

One can compute the resulting action of on and . Unlike in the electromagnetic situation, if is a nonabelian group, then and are not invariant under gauge transformations. Instead, acts on and by conjugating the Lie algebra values.

However, because for , the expressions and in the Lagrangian are invariant under the action of gauge transformations. Thus, provided we transform , we obtain another solution to the Yang–Mills equations.

2.6. Temporal gauge

By applying a gauge transformation, we can set the scalar potential to zero. More precisely, we solve the linear differential equation

for . This gauge transformation sends to .

Restricting to the case , called temporal gauge, we now have

(10)

The Lagrangian becomes

The corresponding Euler–Lagrange equations are

(11)

This is a weak form of the equation

(12)

Setting , we see that is constant by Section 2.4 with . However, when we discretize, we will find the following variational-principle-based proof of this fact more helpful. For all , we have that , so plugging this value of into (11), we find

(13)

Thus, .

In vacuum, both and are the Hodge star , and by taking the Hodge star of (12) and substituting and , we obtain the standard formulation of the time-dependent Yang–Mills equation

2.7. Galerkin semidiscretization

To find numerical solutions to the Yang–Mills equations, we apply Galerkin semidiscretization by restricting the trial functions and test functions in (11) to a finite dimensional subspace . That is, we seek a curve such that

(14)

Here, as in (10), we have , , and we have and .

Unlike the corresponding situation for Maxwell’s equations, (14) is a nonlinear finite-dimensional system of ODEs, since contains the quadratic term and since appears in .

We would like to show that is conserved, at least in some weak sense. We still have that . Thus, . However, showing that vanishes even in a weak sense cannot be done the same way as with Maxwell’s equations.

As in (13), we would like to plug into (14), but the requirement that be in is difficult to satisfy because of the term in . In general, if is a space of piecewise polynomials of degree , then will have degree , so will generally have degree higher than , and thus be an invalid choice of .

As noted by Christiansen and Winther [8], there is a valid choice of , namely, constant -valued functions on , giving us the conservation law

In other words, the total charge on the whole domain is conserved. However, we’d like to have local charge conservation, a much stronger condition.

3. The domain-decomposed Yang–Mills equations

3.1. Domain decomposition

Roughly speaking, the challenge we faced above is that had to be constant, but to get local charge conservation, we needed to be supported on a small region. With domain decomposition, we can resolve this issue by allowing discontinuous test functions. With a discontinuous locally constant , we can get local charge conservation.

We decompose our domain using a triangulation and define discontinuous function spaces with respect to this triangulation.

Definition .

Let

That is, and are discontinuous versions of the spaces and ; the exterior derivatives are only defined after we restrict to a particular element of the triangulation.

Via Lagrange multipliers, we can characterize when a discontinuous form in or is actually “continuous” in the sense of being in or respectively, analogously to how it is done in [6] for scalar fields. We define our spaces of Lagrange multipliers.

Definition .

Let

The level of regularity in these definitions is chosen so that and are well-defined for , , , and via the formula

Each term is in via Hölder’s inequality.

Proposition .

Let . Then if and only if

for all .

Likewise, let . Then if and only if

for all .

Proof.

For , let . Then for , we have

(15)

In particular, if , then this expression is zero as claimed.

Conversely, assume that and that for all . We can define as a distribution on . To show that , let have vanishing trace on . We have, by definition of distributional derivative,

Computing further, using the fact that , , and , we have that

Using Hölder’s inequality, we can bound this expression by

We conclude that the functional is bounded on , so , as desired. We conclude that .

Likewise, assume that and that for all . We define as a distribution on , and in the same way that we computed for , we can compute that for all with vanishing trace, we have

Like we did for , we can bound this expression using the Cauchy–Schwarz inequality.

We conclude that the functional is bounded on , so , as desired. We conclude that .

We’ve shown that and . It remains to show that their traces are zero. For , considering , not necessarily traceless, we have by Equation (15) and the assumption that that for all , so is traceless. ∎

3.2. The domain-decomposed Yang–Mills equations

We now modify the Lagrangian from (7) to allow and to come from the discontinuous function spaces, and we enforce continuity through Lagrange multipliers and . That is, let be a curve in , and let be a curve in . As before, we let and , but in this definition we must compute all derivatives on each element individually, since we do not expect and to have derivatives across the element boundaries.

As before, the regularity assumptions on and imply that and , and so this implies that and . Again, we impose the additional assumption that . Our Lagrangian is now

The Euler–Lagrange equations are then

(16a)