Structure preserving transport stabilized compatible nite element methods for magnetohydrodynamics Golo A. Wimmer1 Xianzhu Tang1

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Structure preserving transport stabilized compatible ﬁnite element methods

for magnetohydrodynamics

Golo A. Wimmer1∗, Xianzhu Tang1

1Los Alamos National Laboratory

Thursday 6th October, 2022

Abstract

We present compatible ﬁnite element space discretizations for the ideal compressible magnetohy-

drodynamic equations. The magnetic ﬁeld is considered both in div- and curl-conforming spaces,

leading to a strongly or weakly preserved zero-divergence condition, respectively. The equations

are discretized in space such that transfers between the kinetic, internal, and magnetic energies

are consistent, leading to a preserved total energy. We also discuss further adjustments to the dis-

cretization required to additionally achieve magnetic helicity preservation. Finally, we describe new

transport stabilization methods for the magnetic ﬁeld equation which maintain the zero-divergence

and energy conservation properties, including one method which also preserves magnetic helicity.

The methods’ preservation and improved stability properties are conﬁrmed numerically using a

steady state and a magnetic dynamo test case.

Keywords. Magnetohydrodynamics, compatible ﬁnite element method, structure preservation, trans-

port stabilization.

1 Introduction

The compressible magnetohydrodynamic (MHD) equations are of interest in a diverse range of

application areas, including astrophysics and magnetic conﬁnement fusion [20]. Their most fun-

damental variant is given by single ﬂuid ideal MHD, where dissipative eﬀects are ignored and all

terms except of the transport one are omitted in Ohm’s law, which deﬁnes the electric ﬁeld to be

used in the magnetic ﬁeld equation [20, 25]. While many physically relevant eﬀects – such as those

related to Hall MHD – are lost in the ideal case, the resulting system of partial diﬀerential equations

still contains a rich structure that is challenging to discretize [19, 25]. This structure in particular

includes the magnetic ﬁeld’s zero-divergence property, and a failure to satisfy this property after

discretization generally leads to large amounts of spurious noise. A wealth of numerical methods

has been developed to avoid this problem, by means of, for example, Lagrange multipliers [38], a

post-processing step [11], 8-wave methods [34], or discretizations that satisfy the zero-divergence

property exactly [16], [22] (and references therein). Another structure is given by a balance be-

tween the kinetic, internal, and magnetic energies, leading to conservation of total energy. In the

context of hydrodynamics, it has long been known that a discrete analogue of this property may

∗correspondence to: gwimmer@lanl.gov

arXiv:2210.02348v1 [math.NA] 5 Oct 2022

lead to more accurate long term predictions [1]. Furthermore, such a discrete analogue may lead

to more accurate results at lower resolutions [43]. Additionally, the ideal MHD equations give rise

to conserved helicities, including the magnetic helicity [21] (and references therein). Finally, next

to these quantities arising from the system’s structure, the MHD equations also contain transport

terms for each of the prognostic variables. When physical dissipative eﬀects are relatively small –

which is often the case for many applications of interest – additional stabilization methods may be

needed to avoid spurious oscillations due to the discrete transport terms.

Recently, a number of compatible ﬁnite element based discretizations have been described for var-

ious forms of the MHD equations, ensuring some or all of the aforementioned structure preserving

properties [18, 21, 22, 26]. In particular, the magnetic ﬁeld’s zero-divergence property is ensured

to hold true via discrete vector calculus identities that are satisﬁed in the compatible ﬁnite ele-

ment approach. Further, energy conservation is achieved by using matching discretizations for the

various terms appearing in the system of equations. This approach may be facilitated by a rich

underlying theory starting from a Lagrangian mechanics point of view [18], or a Hamiltonian one

based on Poisson brackets [30]. Lastly, helicity conservation can be achieved by the use of ap-

propriate projections, moving between the corresponding div- and curl-conforming ﬁnite element

spaces [21]. In the context of compatible ﬁnite element discretizations for hydrodynamics in ocean

and atmosphere modeling, transport stabilization methods have further been incorporated with-

out compromising on energy conservation for the density, thermal, velocity, and vorticity ﬁelds

[6, 15, 28, 31, 43, 42, 41]. However, there is a lack of such methods speciﬁcally aimed at improving

the magnetic ﬁeld transport term’s stability, which comes with the added diﬃculty of maintaining

the latter ﬁeld’s zero-divergence property. One recent approach to resolve this is given by [44],

which concerns general grad-, div-, and curl-conforming convection-diﬀusion problems, and relies

on exponential ﬁtting methods.

In this paper, we introduce two new transport stabilization methods for the div- and curl-conforming

discretizations of the magnetic ﬁeld equation. They are based on interior penalty type formulations

[12, 13], and can be seen as sub-grid resistivity terms. Unlike the addition of a small physical

resistive contribution for stabilization purposes, the latter two methods are consistent with the

ideal MHD equations in the sense that for a continuous current density, the interior penalty terms

vanish. The two methods diﬀer in that the ﬁrst one’s penalty term is analogous to the resistivity

term, while in the second one, the penalty is adjusted to vanish in the computation of the rate of

change of total helicity. Further, we examine a residual based SUPG formulation for the magnetic

ﬁeld equation, which in the context of compatible ﬁnite element methods has been considered for

atmosphere and ocean modeling for the vorticity equation before [6, 41]. All three methods pre-

serve the zero-divergence as well as energy conservation properties. Finally, the three methods’

stabilization properties are studied numerically and compared against each other for both a curl-

and a div-conforming discretization of the magnetic ﬁeld evolution equation.

While magnetic ﬁeld transport constitutes the main focus of the paper, we also introduce new

energy conserving transport stabilization methods for the thermal ﬁeld. Transport stabilized ther-

mohydrodynamic systems have been studied using the compatible ﬁnite element framework before

(e.g. [15] for buoyancy and [43] for potential temperature); however, we found alternative approaches

to ﬁt better into the context of the compressible MHD equations considered here. In particular, we

include an energetically neutral interior penalty term when using the temperature as the thermal

ﬁeld, and an energetically neutral upwind formulation if pressure is used instead.

The remainder of this work is structured as follows: In Section 2, we review the compressible

magnetohydrodynamic equations and their structure preserving properties, as well as the compati-

ble ﬁnite element method. In Section 3, we introduce the structure preserving space discretization,

including stabilization methods for all transport terms. In Section 4, we present and discuss nu-

merical results. Finally, in Section 5, we review our results and discuss ongoing work.

2 Background

In this section, we introduce the compressible magnetohydrodynamic equations, and review some of

its structure preserving properties. Further, we brieﬂy review the compatible ﬁnite element spaces

to be used in the space discretization.

2.1 Compressible MHD

In this work, we derive discretizations for the ideal compressible magnetohydrodynamic equations.

However, in order to motivate the new interior penalty based stabilization methods to be introduced

in the next section, we will start from the resistive MHD equations. They are given by

∂n

∂t +∇ · (nV)=0,(2.1a)

∂V

∂t + (∇ × V)×V+1

2∇|V|2+1

min∇(nT )−1

minj×B= 0,(2.1b)

γ−1∂T

∂t +V· ∇T+nT ∇ · V=η|j|2(2.1c)

∂B

∂t +∇ × E= 0,(2.1d)

E=−V×B+ηj,j=1

µ0∇ × B,(2.1e)

in a domain Ω, and where the prognostic ﬁelds are given by the ion number density n, ion ﬂow

velocity V, plasma temperature T, and magnetic ﬁeld B. Details of the ﬁeld’s underlying spaces

will be postponed to the next section, and in this section, we simply assume suﬃcient continuity for

the diﬀerential operations occurring in the above equations to be well-deﬁned. Note that in view of

this section’s discussion on energy conservation, we wrote the velocity transport term in its vector

invariant form,

(V· ∇)V= (∇ × V)×V+1

2∇|V|2.(2.2)

Additionally, there are two diagnostic ﬁelds, given by the electric ﬁeld E, and the current density j.

The constants miand µ0denote the ion mass and vacuum permeability, respectively, and γ= 5/3.

Finally, ηdenotes the plasma resistivity. If η≡0 – i.e. if there is no resistive or Ohmic heating

eﬀect – then the system of equations reduces to ideal MHD.

Remark 1. Alternatively to the temperature T, it is also possible to consider other thermal

ﬁelds such as the pressure p=nT , with a governing equation given by.

γ−1∂p

∂t +∇ · (pV)+p∇ · V=η|j|2.(2.3)

This formulation leads to a more straightforward description of the system’s energy transfers. How-

ever, here we consider the temperature Tinstead, since we found our resulting discretization (3.1)

to be on average more accurate when compared to our pressure ﬁeld based one (3.13). Further, the

corresponding evolution equation for Tand its discretization are more amenable to incorporate an

anisotropic heat ﬂux – which includes a gradient in Trather than p– and which plays an important

role in, for example, magnetic conﬁnement fusion applications.

Finally, we equip the above equations with free-slip boundary conditions for the velocity ﬁeld,

and mixed boundary conditions for the magnetic ﬁeld. For a domain Ω with a boundary region ∂Ω

and sub-regions ∂Ω1,∂Ω2such that ∂Ω1∪∂Ω2=∂Ω and ∂Ω1∩∂Ω2=∅, these are given by

V·n|∂Ω= 0,(2.4)

B·n|∂Ω1=g1,H×n|∂Ω2=g2,(2.5)

for outward normal unit vector n, and where H:=µ0B. The magnetic ﬁeld boundary conditions

can alternatively also be expressed in terms of E×n=h1and j·n=h2in ∂Ω1and ∂Ω2, respectively

[7]. We note that this may be preferable depending on the choice of ﬁnite element function space,

as will be shown in Section 3.

The above system of equations gives rise to a number of conserved quantities, and in the following,

we will focus on the system’s total energy, total magnetic helicity, and zero-divergence property for

the magnetic ﬁeld. The system’s total energy is given by

H(n, V, T, B) = ZΩ1

2min|V|2+nT

γ−1+|B|2

2µ0dx. (2.6)

Proposition 1a.For boundary conditions of the form V·n|∂Ω= 0 and B·n|∂Ω= 0, the total

energy is conserved in time.

Proof. The rate of change in time of the total energy (2.6) can be computed using the chain rule

dt =δH

δn ,∂n

∂t +δH

δV,∂V

∂t +δH

δT ,∂T

∂t +δH

δB,∂B

∂t ,(2.7)

where – after some abuse of notation1–h., .idenotes the L2inner product, and the variational

derivative related expressions are given by

δH

δn =1

2mi|V|2+T

γ−1,δH

δT =n

γ−1,δH

δV=minV,δH

δB=B

µ0

.(2.8)

1Strictly speaking, the Hamiltonian derivatives are elements of the dual spaces of the respective ﬁelds, and h., .i

denotes a pairing of elements from a space and the space’s dual.

Substituting these expressions and the prognostic ﬁelds’ time derivatives in (2.7) by the evolution

equations (2.1a) - (2.1d), we obtain

δH

δn ,∂n

∂t =−1

2mi|V|2,∇ · (nV)−T

γ−1,∇ · (nV),(2.9a)

δH

δV,∂V

∂t =−hminV,(∇ × V)×Vi − minV,1

2∇|V|2− hV,∇(nT )i+hV,j×Bi,(2.9b)

δH

δT ,∂T

∂t =−n

γ−1,V· ∇T− hnT ∇ · Vi+hηj,ji,(2.9c)

δH

δB,∂B

∂t =−B

µ0

,∇ × E.(2.9d)

The ﬁrst term on the right-hand side of (2.9b) vanishes due to the cross-product of Vwith itself.

Further, upon applying integration by parts for the second and third term on the right-hand side

of (2.9b), as well as the ﬁrst terms on the right-hand sides of (2.9c) and (2.9d), we arrive at

δH

δn ,∂n

∂t =−1

2mi|V|2,∇ · (nV)−T

γ−1,∇ · (nV),(2.10a)

δH

δV,∂V

∂t =1

2mi|V|2,∇ · (nV)+hnT ∇ · Vi+hV,j×Bi,(2.10b)

δH

δT ,∂T

∂t =T

γ−1,∇ · (nV)− hnT ∇ · Vi+hηj,ji,(2.10c)

δH

δB,∂B

∂t =−hj,−V×B+ηji,(2.10d)

where we used the boundary conditions, noting that we used the analogous electric ﬁeld boundary

condition E×n|∂Ω= 0 in place of the magnetic ﬁeld one. Further, we used the deﬁnition of jin

(2.9d) after integrating by parts. Summing over the four terms as in (2.7), we then ﬁnd that all

terms cancel, and the rate of change of Hin time equals zero. 

The computations appearing in the above proof can also be used to uncover the system’s transfers

between the total kinetic, internal, and magnetic energies, which are given by

KE(n, V) = ZΩ

2min|V|2dx, IE(n, T ) = ZΩ

γ−1dx, ME(B) = ZΩ

|B|2

2µ0

dx. (2.11)

Using the variational derivative relations

δH

δn =δKE

δn +δIE

δn ,δH

δT =δIE

δT ,δH

δV=δKE

δV,δH

δB=δME

δB,(2.12)

we then ﬁnd

dKE

dt =−hV,∇(nT )i+hV,j×Bi,(2.13a)

dIE

dt = +hV,∇(nT )i+hηj,ji(2.13b)

dME

dt =−hV,j×Bi−hηj,ji,(2.13c)

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摘要：

StructurepreservingtransportstabilizedcompatibleniteelementmethodsformagnetohydrodynamicsGoloA.Wimmer1*,XianzhuTang11LosAlamosNationalLaboratoryThursday6thOctober,2022AbstractWepresentcompatibleniteelementspacediscretizationsfortheidealcompressiblemagnetohy-drodynamicequations.Themagneticeldiscon...

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Structure preserving transport stabilized compatible nite element methods for magnetohydrodynamics Golo A. Wimmer1 Xianzhu Tang1

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