AN ENERGY STABLE AND MAXIMUM BOUND PRINCIPLE PRESERVING SCHEME FOR THE DYNAMIC GINZBURGLANDAU EQUATIONS UNDER THE TEMPORAL

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AN ENERGY STABLE AND MAXIMUM BOUND PRINCIPLE

PRESERVING SCHEME FOR THE DYNAMIC

GINZBURG–LANDAU EQUATIONS UNDER THE TEMPORAL

GAUGE

LIMIN MA, ZHONGHUA QIAO

Abstract. This paper proposes a decoupled numerical scheme of the time-

dependent Ginzburg–Landau equations under the temporal gauge. For the

magnetic potential and the order parameter, the discrete scheme adopts the

second type Ned´elec element and the linear element for spatial discretization,

respectively; and a linearized backward Euler method and the ﬁrst order ex-

ponential time diﬀerencing method for time discretization, respectively. The

maximum bound principle (MBP) of the order parameter and the energy dis-

sipation law in the discrete sense are proved. The discrete energy stability and

MBP-preservation can guarantee the stability and validity of the numerical

simulations, and further facilitate the adoption of an adaptive time-stepping

strategy, which often plays an important role in long-time simulations of vor-

tex dynamics, especially when the applied magnetic ﬁeld is strong. An optimal

error estimate of the proposed scheme is also given. Numerical examples ver-

ify the theoretical results of the proposed scheme and demonstrate the vortex

motions of superconductors in an external magnetic ﬁeld.

Keywords. Ginzburg–Landau equations, energy stability, maximum bound

principle, error estimate, exponential time diﬀerencing method

AMS subject classiﬁcations. 68Q25, 68R10, 68U05

1. Introduction

In this paper, we consider the transient behavior and vortex motions of supercon-

ductors in an external magnetic ﬁeld Hwhich is described by the time-dependent

Ginzburg–Landau (TDGL) model [20]. This model was ﬁrst established in [21]

with some detailed descriptions in [2, 9, 40]. The TDGL equations in the non-

dimensional form satisfy

(1) 









(∂t+ iκϕ)ψ+i

κ∇+A2

ψ+ (|ψ|2−1)ψ= 0 in Ω ×(0, T ],

σ(∇ϕ+∂tA) + ∇ × (∇ × A) + Re ψ∗(i

κ∇+A)ψ=∇ × Hin Ω ×(0, T ],

with boundary and initial conditions

(2) 





(∇ × A)×n=H×n,(i

κ∇+A)ψ·n= 0 on ∂Ω,

ψ(x, 0) = ψ0(x),A(x, 0) = A0(x) on Ω,

arXiv:2210.11678v2 [math.NA] 25 Jul 2023

2 Limin Ma, and Zhonghua Qiao

where Ω is a bounded domain in Rd(d= 2,3), nis the unit outer normal vec-

tor, the electric potential ϕis a real scalar-valued function, the Ginzburg-Landau

parameter κis an important positive material constant representing the ratio of

penetration length to the coherence length, the relaxation parameter σis a given

positive constant, the magnetic potential Ais a real vector-valued function and

the order parameter ψis a complex scalar-valued function. Physically speaking,

the magnitude of the order parameter |ψ|represents the superconducting density,

where |ψ|= 0 stands for the normal state, |ψ|= 1 for the superconducting state,

and 0 <|ψ|<1 for a mixed state. It is proved in [4] that the order parameter in

the TDGL equations (1) satisﬁes the MBP in the sense that the magnitude of the

order parameter is bounded by 1, i.e.

(3) ∥ψ(·, t)∥∞≤1,∀t > 0

if the initial condition ∥ψ0∥∞≤1. The solution of the corresponding stationary

Ginzburg–Landau equations minimizes the Gibbs energy functional [26, 39]

(4) G(A, ψ) = 1

2∥(i

κ∇+A)ψ∥2

0+1

2∥∇ × A−H∥2

0+1

4∥|ψ|2−1∥2

As analyzed in [34], the energy dissipation law below holds for (1)

(5) d

dtG(A, ψ)≤ −4π(M, ∂tH),

where the magnetization M=1

4π(∇×A−H). Particularly, if the applied magnetic

ﬁeld His stationary, the Gibbs energy of a solution of (1) decreases in time. As

stated in [7], the solution of (1) is not unique, that is given any solution (ψ, A, ϕ), a

gauge transformation Gχ(ψ, A, ϕ)=(ψeiκχ,A+∇χ, ϕ−∂tχ) gives a class of equiv-

alent solutions sharing the same |ψ|and magnetic induction ﬁeld ∇ × A, which are

of physical interests. Although the solutions of (1) under diﬀerent gauges are the-

oretically equivalent, numerical schemes under various gauges are computationally

diﬀerent. The temporal gauge is adopted in the paper since the corresponding

TDGL equations can be viewed as a gradient ﬂow and admits the energy dissipa-

tion property when His stationary. The existence and uniqueness of the TDGL

equations (1)-(2) were given in [4, 7, 31].

For the TDGL equations, some numerical schemes using ﬁnite diﬀerence meth-

ods for spatial discretization were proposed and analyzed to preserve the discrete

MBP and energy bound in [8, 10, 15]. These MBP-preserving ﬁnite diﬀerence

schemes require uniform or rectangular meshes, and the bound of the discrete en-

ergy may be very large in long-time simulations. Numerical schemes using ﬁnite

element methods for spatial discretization can simulate the motion of superconduc-

tors with more general shapes, and are easy to be extended to three-dimensional

simulations. Many ﬁnite element based numerical schemes were proposed and ana-

lyzed for diﬀerent gauges, especially the temporal gauge ϕ= 0 (see e.g., [6, 33, 34])

and the Lorentz gauge ϕ=−∇ · A(see e.g., [3, 16, 18, 27]) under an additional

boundary condition. This boundary condition is indispensable to guarantee the

wellposedness of the discrete problems and analyze the convergence rate of numer-

ical solutions. However, the regularity of the ﬁnite element solution under such

boundary conditions is higher than expected, which leads to some nonphysical phe-

nomena if the mesh is not reﬁned enough. Two mixed ﬁnite element methods using

Hodge decomposition in [28, 30] weakly impose this additional boundary condition

on the approximation of Afor the TDGL equations under the Lorentz gauge, which

An energy stable and MBP-preserving scheme for TDGL equations 3

avoid the nonphysical phenomenon to a certain extent for the TDGL equations in

nonconvex polygons. Recently, a nonlinear numerical scheme with no additional

boundary condition was proposed in [13, 24] for the TDGL equations under the

temporal gauge, which resolves physical-interested phenomena on relative coarse

meshes. The energy dissipation law was proved under a strict restriction on time

steps in [24]. But no MBP analysis was provided for this scheme.

It is of great importance to analyze the MBP (3) and energy dissipation law (5)

for these ﬁnite element based schemes in the literature. Although the discrete MBP

for the TDGL equations is usually observed for ﬁnite element based schemes, it has

not been proved theoretically. The magnitude of the discrete order parameter was

proved to be bounded above in [34] under the assumption τ≲h11

12 and τ≲h2in

two and three dimensions, respectively. The TDGL equations under the Lorentz

gauge cannot be viewed as a gradient ﬂow of the Gibbs energy, and thus the energy

stability analysis of numerical schemes concerning this gauge is diﬃcult and the

relevant work is very limited in the literature. The boundedness of a modiﬁed

energy with an extra term 1

2∥ψ∥2

0was analyzed for the scheme in [31] concerning

the Lorentz gauge with the bound depending on the terminal time. The TDGL

equations under the temporal gauge can be viewed as an L2-gradient ﬂow with

respect to G(A, ψ) and

(6) d

dtG(A, ψ) + ∥∂tA∥2

0+∥∂tψ∥2

0=−4π(M, ∂tH),

which beneﬁts the energy stability analysis of numerical schemes under this par-

ticular gauge. The discrete energy dissipation law was analyzed for the nonlinear

schemes in [6, 24], where the uniqueness of solution for both schemes requires time

step sizes τ≲hd/2where dis the dimension of space. A modiﬁed energy was

proved to be bounded in [34], where the bound tends to inﬁnity as the perturbed

model tends to the original one.

In this paper, we propose a decoupled numerical scheme for the TDGL equations

under the temporal gauge

(7) 









∂tψ+i

κ∇+A2

ψ+ (|ψ|2−1)ψ= 0 in Ω ×(0, T ]

σ∂tA+∇ × (∇ × A) + Re ψ∗(i

κ∇+A)ψ=∇ × Hin Ω ×(0, T ]

with boundary and initial conditions (2). The scheme employs the lowest order

second type Ned´elec element and the linear Lagrange element with mass lumping

for ﬁnite element discretization of Aand ψin space, respectively. For time dis-

cretization, the proposed scheme solves Aﬁrst by the backward Euler method with

the nonlinear term treated explicitly, and then ψby the ﬁrst order exponential time

diﬀerencing (ETD) method [1, 5, 22, 23]. The ETD method has been proved to

preserve the discrete MBP in many applications, see e.g., [11, 12, 25, 29]. Diﬀerent

from the MBP analysis for real-valued diﬀerential equations, the complexity of the

order parameter ψleads to a complex-valued matrix that is not diagonally domi-

nant, and poses diﬃculty in the MBP analysis for (7). Besides, the highly coupled

terms in (7) add to the diﬃculty in analyzing the energy dissipation and error esti-

mate for the proposed decoupled scheme. For the proposed decoupled scheme, we

analyze the discrete MBP-preserving property and the discrete energy dissipation

law with respect to the original Gibbs energy, and give an optimal error estimate.

4 Limin Ma, and Zhonghua Qiao

This is the ﬁrst ﬁnite element based scheme that preserves the strict discrete MBP

(3) theoretically, and the ﬁrst decoupled ﬁnite element based scheme that admits

the discrete energy dissipation law (5) with respect to the original energy (4).

These stabilities are of great beneﬁt since they allow the application of adaptive

time-stepping strategy in [38] to signiﬁcantly speed up long-time simulations.

The rest of the paper is organized as follows. The decoupled numerical scheme

is presented in Section 2. The discrete MBP for the order parameter and an

unconditional energy stability are analyzed in Section 3.1 and Section 3.2, respec-

tively. The error estimate of the numerical scheme is given in Section 4. Some

numerical experiments are carried out in Section 5 to verify the theoretical results

and demonstrate the performance of the proposed scheme in long-time simulations.

The paper ends with some concluding remarks in Section 6.

2. Fully discrete scheme for the TDGL equations

In this section, we present the fully discrete scheme for (7). Some standard

notations are given below. Let Cbe the set of complex numbers, L2(Ω,R), and

H1(Ω,R) be the conventional Sobolev spaces deﬁned on a domain Ω ⊂Rd(d= 2

or 3). For any two complex functions v,w∈L2(Ω,C), denote the L2(Ω,C) inner

product and the norm by (v, w) = RΩvw∗dx,∥v∥2

0=RΩ|v|2dx, respectively, where

w∗is the conjugate of wand |v|is the magnitude of v. Denote the complex-valued

Sobolev space as

H1(Ω,C) = {ϕ=u+ iv:u, v ∈H1(Ω,R)},

and the vector-valued space with dcomponents as

H(curl) = {B:B∈L2(Ω,Rd),∇ × B∈L2(Ω,Rd)}.

The weak formulation of the TDGL equations (7) with boundary conditions (2) is

speciﬁed as follows: ﬁnd (A, ψ)∈H(curl) ×H1(Ω,C) such that

(8) ((σ∂tA,B) + D(ψ;A,B)+(g(ψ),B) = (H,∇ × B),∀B∈H(curl),

(∂tψ, ϕ) + B(A;ψ, ϕ)−(f0(ψ), ϕ)=0,∀ϕ∈H1(Ω,C),

with A(x, 0) = A0(x)∈H(curl) and ψ(x, 0) = ψ0(x)∈H1(Ω,C), where

(9)

D(ψ;A,B)=(∇ × A,∇ × B)+(|ψ|2A,B), g(ψ) = i

2κ(ψ∗∇ψ−ψ∇ψ∗),

B(A;ψ, ϕ) = (( i

κ∇+A)ψ, (i

κ∇+A)ϕ), fµ(x) = (1 − |x|2)x+µx.

Let Thbe a regular partition of Ω, Ehbe the set of all interior edges of Th,te

be the unit tangent vector of an edge e∈ Ehand hKbe the diameter of element

K∈ Th. Deﬁne the mesh size h= maxK∈ThhK. Let P1(K, C) be the set of all

polynomials with degree not greater than one. Deﬁne the linear element space by

Vh={ϕh∈H1(Ω,C)∩C0(Ω,C) : ϕh|K∈P1(K, C)},

and the lowest order second type Ned´elec element space by

Qh={Bh∈H(curl) : Bh|K∈P1(K, R),Ze

Bh·teds is continuous on any e∈ Eh}.

Let ΠLbe the canonical interpolation operator of the linear element, namely

ΠLv(x) = PN

i=1 v(xi)ϕi(x), where Nis the number of vertices {xi}N

i=1 of Th, and

ϕi∈Vhis the corresponding basis function with respect to vertex xiwith ϕi(xj) =

An energy stable and MBP-preserving scheme for TDGL equations 5

δij .Let ωibe the support of ϕi(x). Deﬁne a diagonal matrix D= diag(d1,· · · , dN)

with entries di=|ϕi|0,1,ωi. Denote the inner product (V, W )ℓ2=WHDV =

i=1 ViW∗

i|ϕi|0,1,ωifor any V,W∈CN, and the operators Ih:Vh→CNand

Πh:CN→Vhby Ihw= (w(x1),· · · , w(xN))Tand ΠhW=PN

i=1 Wiϕi(x),respec-

tively. Note that

(10) (Ihv, Ihw)ℓ2= (ΠL(vw∗),1),∥v∥0≲∥Ihv∥ℓ2≲∥v∥0,

where the notation A≲Bmeans that there exists a positive constant C, which

is independent of the mesh size, such that A≤CB. Deﬁne the Ritz projection

RhA∈Qhby

(11) (∇ × (A−RhA),∇ × Bh)+(A−RhA,Bh) = 0,∀Bh∈Qh,

which admits the following estimates on a convex domain [32]:

(12) ∥∇ × (I−Rh)A∥0+∥(I−Rh)A∥0≲h(|A|1+|∇ × A|1),

provided that A,∇ × A∈H1(Ω,Rd) and

(13) h∥∇ × (I−Rh)A∥0+∥(I−Rh)A∥0≲h2|A|2,

provided that A∈H2(Ω,Rd). Given a positive integer Ktand time steps {τi}Kt

i=1,

we divide the time interval by {tn=Pn

i=0 τi: 0 ≤n≤Kt}and T=tKt. For any

function F(·, t), deﬁne Fn=F(·, tn) and ∂n

tF=∂tF(·, tn). For any given sequence

of functions {Fn}, denote dn

tF= (Fn−Fn−1)/τn.

Let A0

h=RhA0and Ψ0

h=Ihψ0. Given the approximation (An−1

h,Ψn−1

h)∈

Qh×CNat the previous time step tn−1, we ﬁrst solve the approximation to An

by applying the backward Euler method for time discretization and treating the

nonlinear terms explicitly. That is to ﬁnd An

h∈Qhsuch that for any Bh∈Qh,

(14) (dn

tAh,Bh) + D(ψn−1

h;An

h,Bh) = (Hn,∇ × Bh)−(g(ψn−1

h),Bh),

where ψn−1

h= ΠhΨn−1

h. We adopt the ﬁrst order exponential time diﬀerencing

method (ETD1) with stabilization for time discretization of ψand the linear ﬁ-

nite element method with mass lumping for spatial discretization by treating the

nonlinear terms B(A;ψ, ϕ) and f0(ψ) in (8) explicitly. To be speciﬁc, we seek

uh∈C1([tn−1, tn]; Vh) such that ψn

h=uh(·, tn)∈Vhwith uh(·, tn−1) = ψn−1

hsuch

that for any ϕh∈Vhand t∈[tn−1, tn],

(ΠL(∂tuhϕ∗

h),1) + B(An

h;uh, ϕh) + µn(ΠL(uhϕ∗

h),1) −(ΠL(fµn(ψn−1

h)ϕ∗

h),1) = 0,

where µn>0 is the stabilization parameter and An

his given by (14). The matrix

form of this formulation reads

(15) 





dtUh(t) = Ln

µn,hUh(t) + fµn(Ψn−1

h),∀t∈[tn−1, tn],

Uh(tn−1)=Ψn−1

where Uh(t) = Ihuh(·, t)∈CNand the entries of the complex matrix Ln

µn,h are

(16) Ln

µn,h =D−1ˆ

Ln−µnI, with ( ˆ

Ln)ij =−B(An

h;ϕj, ϕi).

Since the diagonal matrix Dis positive deﬁnite and the Hermitian matrix ˆ

Lnis

negative semi-deﬁnite, Ln

µn,h is negative deﬁnite for any µn>0, i.e.

(17) W∗Ln

µn,hW≤ −µnW∗W, ∀W∈CN.

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ANENERGYSTABLEANDMAXIMUMBOUNDPRINCIPLEPRESERVINGSCHEMEFORTHEDYNAMICGINZBURG–LANDAUEQUATIONSUNDERTHETEMPORALGAUGELIMINMA,ZHONGHUAQIAOAbstract.Thispaperproposesadecouplednumericalschemeofthetime-dependentGinzburg–Landauequationsunderthetemporalgauge.Forthemagneticpotentialandtheorderparameter,thediscr...

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AN ENERGY STABLE AND MAXIMUM BOUND PRINCIPLE PRESERVING SCHEME FOR THE DYNAMIC GINZBURGLANDAU EQUATIONS UNDER THE TEMPORAL

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