numerical discretization and simulation of ginzburg landau...

Numerical Discretization and Simulation ofGinzburg Landau Models for Superconductivity 1

Alexandre Ardelea, Anand L. Pardhanani, Graham F Carey, andWalter B. RichardsonTexas Institute for Computational and Applied Mathematics, The University of Texas at Austin Austin,

Texas, 78712 USA

Several issues in mathematical modeling and numerical simulation for the time

dependent Ginzburg-Landau equations are investigated. A key point of this study

is the influence of the mesh for the spatial discretization onthe simulation results.

In particular, we demonstrate that a fine mesh resolution in space is necessary, and

that an inadequate mesh resolution will often give rise to spurious solutions which

seem physically correct, but are false. In fact, this appears to be the case for many

published numerical results and there is a general lack of awareness of this mesh

sensitivity issue. Phenomenological studies to examine the effect of the physical

parameters and boundary conditions in 2D and 3D are presented to illustrate the

solution structure and to highlight certain other criticalaspects of the approximation

that affect the numerical solution.

1. INTRODUCTION

Mathematical modeling and simulation have become an essential part of engineering anal-ysis and design in the semiconductor industry, and similar needs are evident for super-conducting microelectronics. Progress in the fabricationof superconductors with criticaltemperatures above 100K, coupled with recent advances in cryocooler technology havecreated opportunities for manufacturing practical superconducting electronic devices [1][2] [3]. The application areas are very broad, ranging from filters for microwave andwireless communications, to junction devices for metrology, sensors and detectors, toSuperconductive Quantum Interference Devices (SQUIDs) asultra-sensitive sensors formicroelectronics and medicine [4] [5]. Particularly attractive for the near term are theRapid Single Flux Quantum Logic (RSFQL) devices for ultra-fast signal processing andsupercomputing [6]. To realize these goals, it is importantto understand the transientbehavior and steady state solution structure for the magnetic flux in superconductors, andto carry out related phenomenological and design studies.Mathematical models for this class of problems exhibit complex solution behavior inthe form of multiple “vortices” that form and evolve to distinct steady state solutionpatterns. Moreover, the dynamical behavior of the solutions to these models is not yetwell understood. These transient and steady state problemsare difficult to approximate1supported by NSF grant 791AT-51067A

numerically and lead to computationally intensive simulations. Many superconductivitysimulation studies in the literature have been directed to approximating the steady stateproblem rather than time-accurate transient simulation. The most widely used approach inengineering and science for solving a steady state problem is to discretize the steady stategoverning equations and then use a nonlinear iterative algorithm such as Newton’s method.This strategy fails if the initial iterate is not sufficiently accurate (e.g., if it is outsidethe domain of attraction of the iterative method). Continuation schemes in a physicalparameter, incremental approximation of boundary conditions or field strength, or othersimilar strategies can alleviate this difficulty but must deal with the complications of turningpoints and bifurcation points. Other approaches such as direct nonlinear descent solutionof the steady problem may also be applicable [7]. Some of the difficulties concerning theconvergence of the nonlinear algorithms are circumvented if, instead, a transient problemis posed and then integrated to a steady state. Moreover, extrapolation of the solution fromthe previous timestep provides a good starting iterate for the next timestep solve, so thissolution scheme is often quite robust. If the transient solution is not of interest, then timeaccuracy is not required and one can elect to construct integration schemes and algorithmsthat have extended domains of stability, and will accelerate the simulation by taking largertimesteps. One can also use continuation ideas within this context to reduce computationtime and can treat the boundary conditions in a similar vein to accelerate convergence tothe steady-state.

These time-integration strategies for steady state solvesbecome even more appealing whenone is also interested in a simulation strategy that can provide time accurate solutions tothe transient problem and perform dynamic sensitivity studies. In this case, it is obviouslypossible to have in one algorithm and simulator, both the time-accurate transient capabilityand the accelerated steady solver. We have developed some ofthe basic ideas underlyingthis approach in previous work [8], [9]. For these reasons, in the present work we considerthe development of an analysis capability and simulation framework for both the transientand steady state problems, using algorithms based on the Time Dependent Ginzburg-Landau (TDGL) model [10], [11]. More importantly, in the present work this enables usto perform fundamental simulation studies of the nonlineardynamics as vortices in theorder parameter form and evolve. This also provides an improved understanding of thesensitivity to boundary conditions and barrier effects at the boundaries.

We discuss in detail the key numerical issues we have investigated, and in particular thosethat concern the sensitivity of simulations to mesh resolution and to domain size. Thesemesh effects have apparently not received adequate attention previously and raise concernsabout the validity of some of the results in the literature. Such issues are becoming espe-cially significant in TCAD, since the design cycle is necessarily of short duration. As thedevice dimensions shrink or new materials and concepts enter the technology, the under-lying physical models become more complex. Thus, the need for fast, accurate, reliablesimulation results using efficient algorithms on high resolution grids is becoming increas-ingly important. We also demonstrate the integration and continuation ideas mentionedabove for accelerated solution to the steady state, as well as for time accurate transientsolution.The remainder of the paper is organized as follows: the physical models, governing equa-tions, boundary and initial conditions are presented in Section 2 ; Section 3 describesthe discretization and integration techniques applied in the later simulation studies; the

2

NUM. DISCRET. AND SIMULATION OF GL MODELS FOR SUPERCONDUCTIVITY 3

numerical results are illustrated and discussed in Section4 and concluding remarks followin Section 5.

2. GINZBURG-LANDAU THEORY AND PHYSICAL MODELS

The behavior of superconductors in a magnetic field and closeto the critical temperature isqualitatively well predicted by the Ginzburg Landau (GL) phenomenological theory [12].This theory combines thermodynamics, electrodynamics andexperimental observations inan attempt to describe the equilibrium state of the superconductor in terms of a space-dependent order parameter and an electromagnetic field (vector potentialA). One ofthe most interesting consequences of the GL theory was discovered by Abrikosov [13]who studied the superconductive properties of metals with short (normal state) mean freepath. He found that, for an infinite superconductor with� (the Ginzburg Landau parameterdefined below)> 1=p2 and external applied fieldsH larger than a critical valueHc1, themagnetic field lines penetrate the superconductor in the form of localized vortices withstrong depletions of the order parameter at their corej j ' 0. He also showed that thesecond order phase transition to the normal state takes place at a valueHc2(> Hc1). Later,it was shown by Gorkov [14], that the GL equations could be derived from the microscopictheory as developed by Bardeen, Cooper and Schrieffer (BCS)[15] for behavior in thevicinity of the critical temperature. Generalizations of the Ginzburg-Landauequations withflux flow and time-dependent order parameter were developed subsequently by Stephenand Suhl [16], Anderson et al. [17], Jakemin and Pike [18], Schmid [10], Gor’kov andElihasberg [11] and others, based on the Gor’kov microscopic formulation [14], [19], [20].Within the GL phenomenological framework, several variants to the basic models havebeen developed to study specific properties of superconductors. This remains an activeresearch topic since the macroscopic models are derived under different assumptions frommicroscopic theory. In fact, one of the important uses of simulation is for comparativevalidation of different models against experiment.In the present work we consider finite domain effects. Of particular interest are thephenomena associated with barrier effects, boundary conditions, and the influence ofdomain size and applied field. From the microscopic description given in [10] and forslowly varying applied fields, the time-dependent GL equations describe the dominantphysics and can be expressed in the form [21], [22], [23] �@ @t + 2ie� �h � = �� j j2 � 14me �i�hr+ 2ec A�2 (1)4��c �1c @A@t +r�� = � 2�emec � ��i�hr+ 2ec A� + �i�hr� 2ec A� ��r�r�A+r�He (2)

Here, (r; t) is the order parameter related to the the density of superconductive electronsandA(r) and�(r) are the vector and scalar potentials respectively.He is the exter-nal applied magnetic field;� and� are temperature dependent coefficients entering theGinzburg-Landau expression for the condensation energy ofthe superconductor [24]; is a

4 A.ARDELEA, A.L.PARDHANANI, G.F.CAREY, AND W.B.RICHARDSON

time relaxation factor;� represents the normal conductivity;me ande are the electron massand electron charge respectively. ForHe < Hc1 the material is fully superconductive withj j = 1 and expels the magnetic flux completely, whereas forHe > Hc2 the superconduc-tive state is destroyed soj j = 0 and the magnetic flux penetrates the material completely.Between these limits, vortices are present in the superconductor. For infinite superconduc-tors and constant applied magnetic field, the steady state equilibrium is characterized byvortices forming a regular triangular pattern. The next lowest energy configuration consistsof a regular square array [13]. During the transient, the vortices migrate and organize intothese patterns as the steady state is approached. For finite domains, a surface barrier effectknown as the Bean-Livingston barrier [25], inhibits vortices from “entering” the domainunlessHc1 < Hbarr < He < Hc2. The evolution and steady state structure of the vortexpatterns for finite domains is a subject of debate and has not been systematically explored.Such effects are best studied within a simulation setting with a time-accurate transientscheme such as the one incorporated in our algorithm. This barrier effect is important tothe behavior and structure of the pattern in the steady state, as well as the stability of thesteady state to "off-design" transient perturbations which are also readily studied with thepresent dual-mode algorithm. The ability to model the boundary layer penetration in thetransient problem and the pattern structure in the steady state case is also dependent on se-lecting the mesh and domain scaling appropriately with respect to the characteristic lengthscales of the problem. Of particular note in the present workis the investigation of meshresolution effects. The quantity� = �=� is known as the the Ginzburg-Landau parameterand represents the ratio of two important characteristic lengths of the model: 1) the co-

herence length� = (�h=4mej�j) 12 which measures the decay length of the order parameter

and 2) the penetration depth� = �2me�c2=16�e2j�j� 12 which corresponds to the depth towhich the magnetic field penetrates the superconductor [24]. Quantities� = =j�j and�A = (me��)=(2e2j�j) can be viewed as relaxation times for the order parameter andforthe current respectively. The particular assumptions madeat the level of the microscopictheory with regard to the nature of the superconductor determine the relationship between�, , �, �, Tc (the critical temperature) and other relevant quantities [10]. In the numericalexperiments, we consider the case of a superconductor surrounded by vacuum. At theboundary, the normal component of the current vanishes and the tangential field is subjectto a Maxwell-type boundary condition such as:(�hir� 2eA) � n = 0 (r�A)� n = He � n (3)

In the main mathematical model that we investigate, the relevant variables are scaled asfollows !s j�j� 0 ; x! � x0 ; t! � t0 ; A! p2Hc�A0 ; �! j�j es �0 (4)

with Hc = Hc2=(p2�) and the other details given in [22], [26]. For the problems consid-ered here, a gauge is specified such that the scalar potential� = 0 in the superconductorbulk domain and the termA � n = 0 in the boundary condition on@ [27], [28]. Oncethe scaling is performed, the TDGL equations take the form [23], [27], [28], [29]:

NUM. DISCRET. AND SIMULATION OF GL MODELS FOR SUPERCONDUCTIVITY 5@ @t = �� i�r�A�2 � j j2 + (5)@A@t = � i2�( �r � r �)� j j2A�r�r�A+r�He (6)

with boundary conditionsr � n = 0 (r�A)� n = He � n (7)

wheren is the exterior unit normal to@. Some particular features of the system Eq.(5)-(7) merit comment: a) is a complex quantity; b) the termr�r �A introduces crossderivatives of the form@xyAi; c) for a uniform field (He = constant) the system is drivenonly through the boundary; and d) the simple form of the natural BC on results from thegauge choice forA on@. Any other gauge choice results in a BC that also includes termsof the form A � n.Another TDGL model applicable to gapless superconductors with paramagnetic impuritiesand based on slightly different microscopic assumptions was proposed by Hu and Thompson[21]. This model further extended the ideas of Gor’kov, Elihasberg [11] and Schmid [10].The scaled version of the corresponding equations then has the form [26], [30], [31].@ @t = 112 h� (�ir�A)2 � j j2 + i (8)@A@t = � i2( �r � r �)� j j2A� �2r�r�A+r� �He (9)

with the same boundary conditions as in Eq.(7). A formal detailed comparison study ofthe models (5)-(6) and (8)-(9) is not the purpose of the present work. The latter model isintroduced in one of the numerical studies to show that the sensitivity of the results to themesh is not unique to the first model, but is of more general concern.

3. NUMERICAL APPROXIMATION

Part of our objective is to develop an approach that is computationally reliable and efficientfor both time-accurate transient simulations and for steady state computations. We areparticularly interested in vortex formation, entry and pattern evolution to the steady state.In addition, we need to explore the importance of the variousmodel parameters, domainsize, applied magnetic field, boundary conditions and modelassumptions.We meet these objectives by a hybrid time-integration approach that can produce timeaccurate transient solutions or stable, lower order time step schemes that permit more rapidacceleration of the computation of the steady state problem. We also consider the useof both stiff and non-stiff integrators. The basic numerical approach is to discretize thePDE systems in space, and to integrate the resulting semidiscrete system from specifiedinitial data. In our 2-dimensional formulation, a mathematical transformation is used to


map the equations from the physical coordinate system to a reference system in which thefinite difference discretization is performed. This makes it possible for our simulator toaccommodate more general physical domains with curvilinear, graded meshes that are stilltopologically structured. This feature will be useful for application problems and situationsin which local grading of the mesh (such as near an interior vortex) is of interest. Thespatial discretization of the mapped equations on the reference domain is performed usinga 9-point finite difference stencil. Another discretization approach based on finite elementsin the physical domain is also being developed, but the particular choice of discretizationscheme (finite difference, finite element or finite volume) isimmaterial to the present work.The grid resolution is, however, a critical point of generalinterest.The spatial discretization of the PDE in the 2D explicit simulations described later isimplemented using a simplified “dial-an-operator” strategy in our research code FDTRANS.The nodal values of the computed quantities and all their derivatives through second orderare made available via a user interface. This approach makesit easy to implement differentphysical models, since the user can construct nodal approximations for any desired PDEsystem (up to second order) using the given solution values and their derivatives. Allspatial derivatives are discretized to second order accuracy in the reference domain, andthe transformation metrics are also discretized to the sameaccuracy.Discretization of the boundary conditions merits a more detailed derivation, since numer-ical experience has shown that their accurate implementation is very important for thisapplication class. With the gauge choice discussed in the previous section, the boundaryconditions become:@ r@n = @ i@n = 0, A � n = 0, and(r �A) � n = He � n. Here rand i respectively denote the real and imaginary components of the order parameter . Inour implementation, derivatives in the boundary conditions are discretized to second orderaccuracy using non-symmetric (one-sided) stencils. However, the edge and corner nodesrequire special treatment for implementing the conditionsonA, since the normal directionis not uniquely defined. To illustrate our approach in 2D, consider the lower-left corner ofthe domain, i.e., (x,y)=(0,0). The two sides of the domain that meet at this corner are they-axis which we denote S1, and the x-axis denoted by S2. If we denoteA = [a; b]T , theconditionA � n = 0 impliesa = 0 on S1 andb = 0 on S2. Using this in the other conditionfor A yields @b@x = Hz on S1, and@a@y = �Hz on S2, whereHz is the z-component of theapplied fieldHe. At the corner we define the normal to be the average of that on S1 and

S2, so the outward unit normal becomes:n = �i�jp2 , wherei, j denote the unit normalvectors in thex�, y� directions respectively. If we use this definition of the normal, thetwo conditions given above forA reduce to the following at the corner node:a+ b = 0 and@b@x � @a@y = Hz. Next we introduce one-sided (second order) difference approximations

for @b@x (along S2) and for@a@y (along S1), and we use the fact thata = 0 on S1 andb = 0on S2. The two boundary conditions then reduce to a2 � 2 linear system which can besolved fora andb. This results in the following conditions at the corner:a = 23 �x�y�x+�yHz,b = � 23 �x�y�x+�yHz. Here we have assumed a uniform mesh with spacing�x, �y in thex�, y� directions. A similar strategy can be used at the other corners. To summarize, weget (a; b) = � f(1;�1); (1; 1); (�1; 1); (�1;�1)g (10)

with � = 23Hz�x�y=(�x+�y) for the corner values beginning at the bottom-left cornerand proceeding in counter-clockwise direction.


Another strategy that we also considered for the corner nodes was to simply seta = b = 0,which seems reasonable since the two sides that intersect ateach corner enforcea = 0 orb = 0 on the other boundary nodes. We remark that these are also theconditions recoveredfrom the above expressions in the asymptotic limit as�x, �y approach0. However,numerical experience shows that these simpler conditions do not work as well on practicalmeshes, and may impact the number of vortices as well as theirbehavior.The spatial discretization step yields an ODE system which is integrated in time with a classof Runge-Kutta schemes. To illustrate the integration strategy, consider the semidiscreteODE system obtained by spatial discretization of the TDGL models described in Section2. The semidiscrete system can be written in the general formdudt = F (u) (11)

whereu(t) is the solution vector of grid point unknowns. Integrating through one step�tnfromu(tn) tou(tn +�tn), the Runge-Kutta schemes that we utilize can be written in thefollowing general form [32] un+1 = un +�t qXi=1 biki (12)

where the number of stagesq, the value of the coefficientsbi, and the form of the functionski depend upon the specific scheme under consideration. Both explicit and implicit RKschemes can be accommodated in the above formulation. For example, in the case ofexplicit schemes the functionski have the formki = F(un +�t i�1Xj=1 aijkj) (13)

where the coefficientsaij constitute a strictly lower triangularq � q matrix whose entriesdepend upon the specific integration scheme. Since the scheme is explicit, eachki dependsonly upon the previously computed stages.Explicit integration methods described by Eq.(13) are of special interest for parallel solution,which is important for this application class due to the meshresolution requirements, aswe shall see later. Multistage explicit Runge-Kutta schemes are usually constructed tomaximize the order of accuracy for a given number of stages. Such schemes typicallyhave relatively small domains of stability. However, it is possible to construct alternativeschemes that are of lower order accuracy but with larger stability regions permitting largertimesteps with the same number of stages. This can be quite effective for time accuratesolution of moderately stiff problems when it is combined with an adaptive timesteppingstrategy. However, this type of approach may also be used to accelerate the convergenceof time-stepping schemes to steady-state even in the case where the solution is no longerrequired to maintain time accuracy. In this latter case, onemay even violate the stabilityrestriction occasionally during the procedure since the timestepping algorithm functionsessentially as an iterative method with each successive timestep solution now correspondingto an “iterate.” Of course, this latter strategy is not time accurate, so the intermediate iteratesno longer have any relation to the true time evolution of the solution. Moreover, one canadapt the choice of parameters as the solution evolves to meet the needs of the simulation


goal. More specifically,q-stage schemes such as Eq.(12) and Eq.(13) can be constructedto exhibit different accuracy and stability properties by choosing different values for thecoefficientsbi andaij . For example, the following coefficient values define a common4-stage, fourth order accurate schemea21 = a32 = 0:5; a43 = 1; aij = 0 otherwiseb1 = b4 = 1=6; b2 = b3 = 1=3 (14)

It can be shown through linear stability analysis for the standard model problem that thisscheme is stable forj��tj < 2:78 where� is the largest eigenvalue of the ODE system(here we assume� to be real for simplicity). On the other hand, coefficient values foranother 4-stage scheme which is only first order accurate are[33]a21 = 0:0156; a32 = 0:05; a43 = 0:1562; aij = 0 otherwiseb1 = b2 = b3 = 0; b4 = 1 (15)

This scheme can be shown to be stable forj��tj < 32 (again� is assumed to be real). Theper-step cost of using the schemes (13) and (14) is the same, since each involves the samenumber of stages and function evaluations. Thus, the total cost of a simulation would dependupon the total number of steps required by each scheme. In general, when stiffness is not asignificant factor and step-size selection is based on a specified accuracy requirement onewould expect the higher order method to yield higher computational efficiency. Dependingon the severity of stiffness, however, the lower order method may perform more efficiently,particularly if it is more stable or costs less per step. Thisis because, as stiffness increases,step-size selection may become dominated by stability constraints rather than accuracyrequirements. In the numerical studies described in Subsection 4.3, we compare these 4-stage schemes and present several simulation cases where the lower order explicit methodperforms significantly better due to its greater stability.Exploratory work has also been carried out in 3D for this superconductivity application. Inthese simulations, the problem is solved using the PEPPER3 [34] semiconductor processmodeling software. The PDEs are again discretized in space to obtain a semidiscrete ODEsystem of the form (10). In this instance, a seven-point FD stencil for the Laplacian isused with corresponding discretizations for the mixed partial derivatives. PEPPER3 callsthe system integrator CVODE, a variable-step, variable-order integrator which uses eithera Backward Differentiation Formula (BDF) or Adams method. These strategies follow thelinear multistep formula,K1Xi=0 �n;i un�i + �tn K2Xi=0 �n;iF (un�i) = 0 (16)

where�tn is the stepsize,�n;0 = �1, and the coefficients are chosen such that theformulas are exact for a polynomial of degreeq. The Adams-Moulton formula usesK1 = 1, K2 = q, and is particularly effective on nonstiff problems. BDF methodsfollow by takingK1 = q;K2 = 0. For both integration schemes, the resulting algebraicsystem is solved using a quasi-Newton iteration. This increases the work per time stepas compared to a fixed-point iteration, but for stiff problems this is offset by the betterconvergence properties enjoyed by Newton’s method. A GMRESscheme is used to solve


the linearized problem. The predictor-corrector integration strategy selects both the orderof the method and stepsize to control the vector of estimatedlocal errore = utrue � u,so that((1=N)PNi=1 (ei=[�juij+ �])2)1=2 � 1, where� and� are relative and absolutetolerances, respectively.

4. RESULTS AND DISCUSSION

The following numerical studies strongly suggest that the discretized TDGL equations in-corporate certain distinctive numerical characteristicsthat have not yet received adequateattention in the literature. These include effects associated with the mesh resolution, phys-ical domain size and characteristic length scales, boundary condition treatments, stiffnessand choice of integration schemes. Our results reveal several issues that are critical to amore complete understanding of this problem. We report results from these numerical ex-periments in the discussions that follows. For the 2D studies described belowHe = Heezwith He constant. The square domain (2D problem) is discretized with a uniform rectan-gular mesh of equal spacing in both directions. Unless otherwise indicated, we use� = 2in all the cases, andA = 0 in at t = 0.The early transient behavior was first examined for the boundary conditions introduced inSection 2. We considered several cases with different initial data for including randomdata. In all cases, the approximate solution evolves rapidly (within 2' 5 time units) to aform wherej j � 1 throughout the domain and no vortices are present. Hence, for the timeaccurate solutions, to replicate our results one may take a randomized field with j j = 1(e.g.,Re( ) random, andIm( ) = (1�Re2( ))1=2). In the following numerical results,the vortices are displayed by plotting the magnitude of over the simulation domain. Asseen in the plot sequences for the time accurate solution, vortices enter the domain at theboundary and slowly move into the interior, pushed by the Lorentz-like force from theMeissner current. Eventually they form a stationary pattern in the interior, which is thensustained indefinitely. Figure 1 shows this evolution for a representative problem on asquare domain of size24� � 24� and applied fieldHe = 0:95. In our scaling� = 1, sowith � = 2 this domain corresponds to a square of size12 � 12 in scaled units, and theapplied field translates toHe = 0:475Hc2. Note the symmetry of the vortex penetrationat the earlier times shown in (a) and (b). This kind of transient configuration has beenreported in the literature much more frequently than steadystate results. The time takenfor all the vortices to enter the domain is generally much shorter than the time neededby these vortices to rearrange and find a stable steady state configuration. For the timeaccurate regime we specify an absolute error tolerance of10�5 in the 2D simulations withthe RK scheme. It takest ' 80 units of time to obtain the configuration in Figure 1 (b) butt ' 1000 is needed to get the vortex lattice in Figure 1 (d). Similarly, the result later inFigure 8 (a) is obtained aftert ' 50 but one has to integrate up tot ' 500 to obtain Figure8 (b). If the transient is not of interest, then one can accelerate the timestepping strategyas suggested earlier or employ other strategies such as the continuation scheme applied inSubsection 4.4.For convenience we organize the remainder of this discussion into the following subsections:(1) Mesh sensitivity studies; (2) Effect of domain size and applied field; (3) Stiffness andintegration methods; and (4) Three dimensional studies andother extensions.


4.1. Mesh sensitivity studies

The key issue that we explore here is the influence of mesh resolution on the qualitativefeatures of the computed result. Our goal is to demonstrate that inadequate resolution maylead to spurious approximate solutions. It may be very difficult to visually recognize thatsuch an approximate solution is erroneous since it converges on the chosen mesh and mayappear physically reasonable. We define a “spurious approximate solution” in the presentcontext as one whose qualitative features are substantially different from the “correct”approximate solution obtained using much finer meshes for the same problem on the samecomputational domain.

In the first test case Eq.(7)-(9) is approximated on a12��12� domain with an applied fieldHe = 0:44Hc2. The scaling used in this system is based onx! � x0 (instead ofx! � x0used in Eq.(5)-(7)). Thus, with� = 2, � = 2, the scaled length of each side of the domainis 12 units. Three uniform grids of size32� 32, 64� 64, and128� 128 were used. Thesimulation was carried out with the same integration schemeand error tolerance, throught = 500 units of time. Figure 2 (a)-(c) show contour plots of the magnitude of the orderparameter. For the32 � 32 mesh, no vortex enters the domain, whereas for the64 � 64and128� 128 mesh four vortices are seen!

For the second case, the problem Eq.(5)-(7) withHe = 0:95 = 0:475Hc2 is discretized ona24��24� domain with four grid densities: (a)64�64, (b)90�90, (c)128�128 and (d)256�256grid points. The results are shown in Figure 3 for simulations on these respectivegrids. For the coarsest grid (a), vortices penetrate the domain and start to redistributenormally. The fact that the mesh resolution is inadequate appears only later during thetransient calculation: att ' 180, the order parameter starts to exhibit extended regions ofdepletion (j j = 0) which are non-physical. These regions increase at a very slow rate andFigure 3 (a) shows the result att = 400. If the integration is continued, the trivial solution(j j = 0) is approached everywhere. This artifact of poor mesh resolution disappears whenthe problem is recomputed on finer grids! This result on the coarsest mesh is a strikingdeparture from the nontrivial solutions obtained on the finer grids as indicated in Figures3 (b) - (d) where the integration has been carried out until the solution remains unchangedfor� 103 units of time. So the solution is changing very slowly, and for practical purposesa steady state has essentially been reached. Note that thesenontrivial steady states alsodiffer.

Hence, in the case of Figure 3 we find that, for a number of intermediate grid densitiesranging from90� 90 to approximately200� 200, the stationary result is grid dependent:denser grids allow more vortices to penetrate and to rearrange in a more regular distributionthan the coarser grids permit. The numerical results strongly suggest that the pattern shownin Figure 3 (d) is the correct answer because it remains unchanged under further meshrefinement for meshes between' 200� 200 and324� 324. One may argue that, giventhe nature of the problem (symmetries in domain geometry andboundary conditions), theintermediate results (b) and (c) look suspicious. However,for a more general problemdomain and boundary conditions, it will be very difficult to reject false numerical solutionsor suspect their validity without further analysis or grid refinement studies. An errorindicator analysis similar to those applied in adaptive mesh strategies would be useful[35]. Otherwise, as applied here, one must at least recompute on successively refinedgrids to verify whether a computed result is physically correct and that it yields the sameconfiguration with the same number of vortices. In this case the difference in successive


solutions is the error indicator and is, of course, more reliable than a residual or truncationerror based estimate. This can also be augmented by Richardson extrapolation.

We would like to point out that we have carried out several very long term simulations in aneffort to verify that the square lattice we see in Figure 3 (d)in fact represents the steady-stateresult. For example, we recomputed the results shown in Figure 3 (d) up to t=5000 andsaw no change between t=1000 and t=5000. The square latticesin our steady-state resultsoccur instead of the theoretical Abrikosov triangular lattices due to the the effects of theboundary and also the specifics of the problem domain size andapplied field. We explorethese points in greater detail next.

According to this theory, if the domain is infinite the lattice is triangular,but as we now show,the situation is far more complex than the known theoreticalresults suggest. In the finitedomain case, domain size, applied field and boundary effectsplay a significant role in boththe evolution of the vortex patterns and their final structure. We develop these ideas nextwith a series of supporting numerical investigations. We begin with the problem domainand model shown in Figure 3, but with varying applied field. Let us consider the situationwhen the applied field is incrementally increased fromHe = 0:78 = 0:39Hc2. For thisand lower field values, the numerical results show that no vortices enter the domain. Weperformed these calculations using a200� 200mesh. Next,He is increased incrementallyand the calculations repeated from the same initial data andwith 200 � 200 mesh. AtHe = 0:80 = 0:40Hc2 andt � 60, we see four vortices simultaneously begin to enter thedomain at the mid-side locations (see Fig. 4 (a)), and have fully entered byt � 70 (seeFig. 4 (b)). They continue to move toward the center (Fig. 4 (c)) and then rotate intothe steady-state square configuration in Figure 4 (d). Note that a square pattern, and not atriangular pattern, is obtained in the steady-state.

A series of similar calculations were made with applied fieldincreased by incrementsthroughHe � 1:02 = 0:51hc2. For field values aboveHe � 0:87 we also repeated thecalculations with a mesh resolution of300 � 300 to verify that we get the same numberof vortices as that on the200 � 200 mesh. We find that in the rangeHe � 0:80 � 0:82four vortices enter the domain and evolve to a square pattern. The key difference is thatasHe increases, the vortices enter the domain earlier; for example, atHe = 0:81 theybegin to enter byt � 40, and are fully inside the domain byt � 50. As the field valueincreases toHe � 0:82 we see eight vortices enter the domain. ForHe = 0:82 they beginto enter att � 30 in the form of pairs from the mid-sides of the domain. They then evolveinto a pattern that is not a square (since 8 is not a perfect square). In this case, one candiscern subgroups of triples that suggest a “triangular” lattice of sorts. Figure 5 shows thisevolution sequence for the caseHe = 0:84. As we continue to increase the applied field,the basic behavior with the eight vortices follows that observed earlier for the four vortexcase, until atHe � 0:88 we find that an additional vortex begins to enter at each mid-side(behind the first pair) yielding twelve vortices. These twelve vortices evolve to a patternthat also resembles a “triangular” sort of structure (see Fig. 6). Similar 12-vortex solutionsare seen for field values in the rangeHe � 0:88 � 0:93, again the key difference beingthat larger field values cause the vortices to enter the domain at lower values oft. Figure 6shows the 12-vortex evolution sequence for an applied fieldHe = 0:90. Notice that we arenow using a300� 300 mesh. Continuing this experiment, asHe increases further we see16 vortices enter the domain in the rangeHe � 0:94� 0:99. Since 16 is a perfect square,the vortex pattern in these cases does evolve to a square lattice in the long-term (see Fig.


3). If we continue to increaseHe, we see 20 vortices forHe � 1:0, and they evolve to anirregular pattern, since 20 is not a perfect square.In addition to the systematic experiment discussed above, we have also computed resultsfor certain selected field values that are larger thanHe = 1:02 which are shown later. Asexpected, in these cases we see even larger number of vortices, and they evolve in wayssimilar to those seen in the preceding experiment. More comprehensive results and theirdetailed discussion is given in [36].One of the key conclusions from this experiment is that the final configuration of the vortexpattern depends upon the number of vortices admitted into the domain. It may also dependupon the shape of the domain, which is a square in our studies.We have made an extensive literature study of computationalresults pertaining to this meshsensitivity problem for GL simulations, and have found thatthe number of grid points percoherence length� or the grid density typically varies between 2 to 5 [26], [28], [30].Other studies [29], [31], [37] failed to provide the information necessary to determinethe grid density (either the mesh size or the physical domainsize were not available) sopresumably these are not felt to be important.For our preceding simulation studies, the grid densities (i.e., nodes per coherence length)were 2.67 for Figure 2 (a) and Figure 3 (a), 5.33 for Figure 2 (b) and Figure 3 (c), 10.66 forFigure 2 (c) and Figure 3 (d), and as high as 12.66 for Figure 7 (c). Note that meshes withless than 7-10 grid points per� generate spurious results so it is reasonable to infer thatmany results in the literature are suspect, especially in the absence of a mesh refinementstudy. The required mesh density numbers vary withHe; the higher the applied field, thegreater the requirement for fine meshes becomes.The mesh sensitivity behavior in this application is very different and more critical thanthat seen in many other application areas in engineering or science. Significantly, it isquite different in nature than the mesh behavior observed insemiconductor simulation formicroelectronic devices [38]. In device applications,poor resolution may lead to divergenceand breakdown of the numerical algorithm. This is often the case in applications wheremesh sensitivity stems from convective effects. However, in such cases, with the useof certain special stabilization or upwind strategies, it is possible to realize at least aqualitatively correct solution even on coarse meshes. Thismay lead to an overly dissipativeapproximationbut there is little risk of converging to a radically different solution of the typewhich was exhibited here. In the present application we haveseen that spurious solutionsare pervasive and often easy to mistake for the true solution. This would be particularlydetrimental for the designer who would use the results of a numerical simulation to modelthe vortex dynamics and vortex pinning in a superconductivematerial [39]. Therefore, itis imperative that mesh refinement studies and reliability analysis be performed to confirmthe validity of all numerical results.

4.2. Effect of domain sizeAs the previous section demonstrates, the applied field is a key factor affecting the structureof the vortices and their arrangement. As the magnitude of the field increases, the averageseparation between vortices decreases or the number vortices within a given simulationdomain increases. As also noted there, the mesh must be refined as the field is progressivelyincreased and the mesh scaling with the field is not linear. Thus, whereas a128� 128 gridappears sufficient to accommodate the lower applied fieldHe = 0:80, very fine grids like


the 304 � 304 mesh are required for the large applied fieldHe = 1:1. In addition, thebifurcation points at which the vortex pattern increases innumber are not equally separated.Next we consider the effect of domain size, since we have found that the number of vorticesentering the domain is also a very strong function of its size. We repeated the previousexperiment with a domain size of16� 16 scaled units (i.e.,32�� 32� for our scaling), forcertain selected values of applied field. For example, if we consider a field ofHe = 0:90for this domain size, the number of vortices increases to 28 (from 12 for our previous caseof a 12� 12 domain). In addition, the mesh resolution required to obtain a valid solutionbecomes even finer. With the smaller domain we could use a200 � 200 mesh, whichcorresponds to a spacing of0:06 in each direction. This same spacing for the larger domainappears inadequate, and we have to use at least300� 300 nodes to get valid solutions.The reason for this behavior is that on finite simulation domains, the arrangement of thevortices is influenced by the surface barrier effect. If the physical dimensions of the domainare of the order of a few coherence lengths e.g.' 10�, then the vortex dynamics is stronglyaffected by the presence of the boundary.For example, at a higher fieldHe = 1:10, there are four vortices in the10� � 10� domainand only one in the8� � 8� domain. For a lower field withHe = 0:88 four vorticesare present in a12� � 12� domain as seen in Figure 8, but there are no vortices in asmaller10� � 10� domain. Figure 8 shows that the four vortices which are present in the12��12� domain att = 160 (a) start a rotation at aroundt ' 300 and end up in the lockedconfiguration (b). This configuration is largely determinedby the domain geometry andboundary proximity.From these results it can be deduced that the number of vortices and their relative locationdepend upon both domain size and applied field. We emphasize this point because theinfluence of domain size apparently has not been explored in previous investigations. Sim-ulations on large physical domains with fine resolution, through lengthy transients, wouldfurther impact computational resource needs and algorithmdesign.

4.3. Stiffness and integration methodsIn this application stiffness becomes increasingly significant as the mesh resolution becomesfiner. The discretized diffusion operator contributes to the stiffness and so do some of theother terms, which have strong nonlinearities and involve cross-derivatives. Numericalexperience confirms the expected increase of stiffness as the mesh is refined. This is seen inthe form of decreasing integration step sizes and reduced effectiveness of adaptive step-sizecontrol algorithms. In our 2D simulator, the step-size control algorithm keeps track of thenumber of “rejected steps,” and this number increases (relative to the total number of steps)as stiffness increases.In Section 3 we briefly discussed integration methods and ourreasons for exploring a new,more stable, class of explicit methods. We now present numerical results that compare theperformance of the 4-stage, 4-th order and the 4-stage, 1-storder Runge-Kutta schemesgiven in equations (13) and (14). In all the test cases we specify a fixed temporal errortolerance, which the integration algorithm satisfies at each step by adaptively selectingthe step-size. Numerical tests have been performed for various mesh resolutions andfield values. The results of one such study are shown in Table 1, which compares thecomputational performance of the 1st and 4th order, 4-stageintegration methods. Herewe consider again the square domain of size24�, with He = 0:88 and with a specified


time truncation error tolerance of10�5. The three rows in the table correspond to differentgrid resolutions but not the same problem. The 1st order scheme performs better thanthe 4th order scheme in all cases, mainly because of the higher stability of this particular1st order scheme, which can use much larger step-sizes for these stiff problems. Remark:in the above simulations the lower order scheme is still timeaccurate and as noted inthe introduction, one could take this further by using stepsthat are much larger than thatrequired for time accuracy to further speed up the computation to a steady state solution.We have not used this last approach in the present studies.

4.4. Three dimensional studies and other extensionsResults for the 3D problem have been reported in the literature [39], using supercomputers,the explicit Euler method, and hundreds of thousands of timesteps. One of our goals isto develop a robust 3D code that will run in reasonable time ona workstation. Considerthe problem of timestepping to the steady state, without requiring time-accurate solution.Then various continuation schemes can be introduced to accelerate convergence of thealgorithm to the steady state. For example, one can elect to apply incremental continuationin a parameter, modify the coefficients in the time-dependent term, or modify the boundaryconditions.In the present case, the boundary conditions for the time dependent simulation are modifiedto accelerate “arrival” to the correct steady state. Assumethat the steady state solution is tobe computed with a mixed condition such asB(u) � �@u=@n+�u� = 0. In the presentwork this is implemented by considering the least-squares functional�(u) = � 12kB(u)k2and the associated ODEdz=dt = �r�(z(t)), describing the path of steepest descent.Here� is a factor that determines the weight given to enforcing theboundary conditionrelative to satisfying the PDE system. This can be done in thefunction space setting(H� 12 ) or in the semidiscrete setting where it gives rise to another differential equationfor each boundary point, and is a convenient continuation process for gradually enforcingthe boundary conditions throughout the integration, instead of forcing them to be satisfiedat every timestep. This gives a first-order ODE at boundary nodes for the components ofA, which is much more amenable to the multistep integration strategy than algebraicallyenforcing the flux BC’s.In the tests described here, meshes are uniform in each direction and the error controlis based on the above criteria, with relative tolerance� = 10�2 and absolute tolerance� = 10�5. Table 2 compares the Adams and BDF integration strategies for TDGL on asequence of meshes. The physical parameters are� = 2, t = 100,He = (0:1; 0:1; 1), and = 6:67�6:67�3:34 in � units. In all cases the Adams method was between two to threetimes faster than BDF. Moreover, the order for the Adams scheme started at 1, increasedto 3, and then settled down to 2 for most of the integration, whereas BDF used primarilyfirst order. If the tolerances were to be reduced significantly, BDF should compare morefavorably with Adams.The left panel of Figure 9 shows the steady state results with� = 2, a mesh of50�50�25,andHe = (0:3; 0:3; 1). Isosurfacesj j = 0:3 are plotted att = 100. Eight well-definedvortices with slight tilts due toHe 6= e3 are evident. The right panel of Figure 9 shows theresults withHe = (0:6; 0:6; 1). The tilt becomes much more extreme, so that the vorticesleave the domain through its sides rather than the top. A finermesh in thez-direction isnecessary to properly resolve the normal intersection of the vortex axis with the side wall.


As mesh resolution is increased, the stiffness of the ODE system increases, causing anexponential increase in cpu time. Fort = 50, He = (0:6; 0:6; 1), and grid80� 80� 40,the Adams method required 15169 time steps and took 315,365 cpu seconds on a 500MHZ processor. There are several approaches to alleviate this performance degradation.One could employ a multigrid strategy for the linear solver.More generally, variouspreconditioners could be tried within the CVODE framework.For example, Sobolevgradient preconditioning within a descent algorithm has been successfully applied to the2D stationary problem [42].As regards the sensitivity of the approximate solution to the mesh resolution, results similarto those described above for the 2D problem were obtained here. As the mesh is refinedfrom30�30�25 to50�50�25, the number of vortices that the mesh can support increasesfrom four to eight. The need for sufficient spatial resolution to insure an accurate solutionis consistent with theoretical analyses [40] on a related complex-valued Ginzburg-Landauequation,�� = �2 (1 � j j2) in , = g on @ whereg : @ ! S1 is a givenfunction mapping onto the unit circle. Ifg has topological degreed > 0, then this vorticitywill result in the formation ofd vortices. To the extent that this equation is an analogue ofTDGL, one would expect that, until the mesh is sufficiently refined to accurately capturethis vorticity information, vortices will be present but insmaller numbers.

5. CONCLUDING REMARKS

Our numerical studies for transient and steady state solution of the time-dependent Ginzburg-Landau model for superconductivity reveal several interesting issues. Foremost amongthese is mesh sensitivity and the distinctive ways in which it is manifest. Our resultsdemonstrate that inadequate mesh resolution often causes the numerical results to exhibitspurious, non-physical solutions. Furthermore, in many cases such solutions appear physi-cally very reasonable, which makes it easy to mistake them for a valid solution. This wouldbe particularly true on domains of irregular shape or with boundary conditions that are lesssymmetric. Thus, it is important to validate any simulationresult for this problem classby recomputing it on finer meshes until its features remain invariant with respect to meshdensity and through the use of reliable computed error indicators.The need for sufficient spatial resolution to insure an accurate solution is consistent withtheoretical analyses by Bethuel, Brezis, and Helein [40] ona related complex-valuedGinzburg-Landau equation,�� = �2 (1�j j2) in, = g on@, whereg : @! S1is a given function mapping into the unit circle. They have shown that if g has sometopological degreed > 0, then this vorticity will result in the formation ofd vortices. Tothe extent that the Bethuel, Brezis,and Helein problem is ananalogue of (Eq. (5)-(7)), onewould expect that until the mesh is sufficiently refined to accurately capture this vorticityinformation, vortices will be present but in smaller numbers.A related issue is that the domain size necessary to realize the essential solution featuresmust exceed a minimal number of characteristic lengths. This fact, coupled with therelatively fine resolution needed, implies that meshes withvery large number of nodes areoften required. Thus, computational efficiency (in terms oftime and memory cost) is acrucial requirement for practical simulations over the lengthy transients that we typicallysee. A hybrid approach is explored here that is applicable toboth the transient and steadystate simulations. In the latter case we may even elect to have a scheme that is no longer timeaccurate and in this manner accelerate "convergence" of theresulting "iterative" methodto a steady state. Similarly, we can combine continuation ina parameter or otherwise as


indicated in the 3D simulation with the modified boundary condition. More generally, for atime accurate solution, the large problem sizes would suggest that explicit methods might bepreferable. On the other hand, another consequence of the fine spatial resolution is higherstiffness in the discrete systems, which degrades the efficiency of explicit schemes. Thus,schemes that offer high stability would clearly be advantageous for this application class.We have explored such a class of explicit Runge-Kutta schemes with adaptive timestepping.The best strategy, however, may well be a hybrid integrationscheme that is explicit in theinitial stages of the simulation, which is dominated by rapid transients, and later switchesto an implicit scheme after the transients decay and stability becomes the primary limitingfactor and vortices are migrating slowly toward a steady state configuration.Exploratory three dimensional studies have also been presented which show that manyof the numerical issues we see in 2D become even more criticalin 3D. In particular,mesh resolution and domain size requirements cause computational efficiency to become aparamount concern.Finally, we remark that, in addition to algorithmic improvements that lead to higher effi-ciency, we are also exploring distributed parallel implementation on computer clusters togain speedup in simulation time. Preliminary work in 2D has been very promising, and weplan to extend these ideas to 3D.


REFERENCES

1. Proceedings of the Fourth Symposium on Low Temperature Electronics and High Temperature Superconduc-tivity / Symposium on Low Temperature Electronics and High Temperature Superconductivity (4th : 1997 :Montreal, Quebec) / Pennington, NJ / 1997.

2. Proceedings of EUCAS 1997, The Third European Conferenceon Applied Superconductivity held in the Neth/ European Conference on Applied Superconductivity (3rd : 1997 : Veldhoven, Netherlands) / Bristol / 1997.

3. Proceedings of the 1999 University of Miami Conference onHigh Temperature Superconductivity, CoralGables, January 7-13,1999 - Woodbury, N.Y. : American Institute of Physics, (1999).

4. T. Van Duzer and C. W. Turner, Superconductive Devices andCircuits, Prentice Hall PTR, Upper SaddleRiver, New Jersey, 1999, ISBN 0-13-262742-6.

5. T.P.Orlando, K.A.Delin,Foundations of Applied Superconductivity, (Addison-Wesley,1991).

6. Proceedings of the 6th International Conference on Materials and High Temperature Superconductivity, Feb.20-25 2000, Houston, Texas (2000) : to be published.

7. J. W. Neuberger,Sobolev Gradients and Differential Equations, (Springer V., 1997).

8. A.A.Lorber, G.F.Carey, W.D.Joubert, ODE Recursions andIterative Solvers for Linear Equations,SIAM J.Sci. Comput.17, 1, p.65 (1996).

9. A.A.Lorber, G.F.Carey, S.W.Bova, C.Harle, AcceleratedSolution of Non-Linear Flow Problems Using Cheby-shev Iteration Polynomial-Based Runge-Kutta Recursions,Comp. and Fluids, 27, 7, p.769 (1998).

10. A. Schmid, A time-dependent Ginzburg-Landau Equation and its application to the problem of resistivity inthe mixed state,Phys. Kondens Matter.5, p.302 (1966).

11. L. P. Gor’kov, G. M. Eliashberg, Generalization of the Ginzburg-Landau equations for non-stationnaryproblems in the case of alloys with paramagnetic impurities, Sov. Phys. JETP27, 2, p.328 (1968).

12. V.L.Ginzburg, L.D.Landau,J. Exptl. Theoret. Phys.(USSR),20, p.1064 (1950).

13. A. A. Abrikosov,Soviet Phys. - JETP.5, p.1174 (1957).

14. L. P. Gor’kov, Microscopic Derivation of the Ginzburg-Landau equations in the theory of superconductivity,Soviet Physics - JETP9, 6, p.1364 (1959).

15. J. Bardeen, L N. Cooper, J. R. Schrieffer,Phys. Rev.108, p.1175 (1957).

16. M. J. Stephen, H. Suhl, Weak Time Dependence in Pure Superconductors,Phys. Rev. Lett.13, 26, p.797(1964).

17. P. W. Anderson, N. R. Werthamer, J. M. Luttinger, An Additional Equation in the Phenomenology ofSuperconductivity: Resistive Effects,Phys. Rev., 138, 4A, p.1157 (1965).

18. E. Jakemin, E. R. Pike, Time-Dependent Ginsburg-LandauTheory,Phys. Lett., 20, 6, p.593 (1966).

19. L. P. Gor’kov, On the Energy Spectrum of Superconductors, Soviet Phys. JETP, 34, 3, p.505 (1958).

20. A. A. Abrikosov, L. P. Gork’ov, I. E. Dzyaloshinski,Methods of Quantum Field Theory in Statistical Mechanicstranslated by R. A. Silverman, (Prentice-Hall Inc., Englewood Cliffs, New jersey, Chap.7, 1963).

21. C. R. Hu, R. S. Thompson, Dynamic Structure of vortices inSuperconductors IIH << Hc2, Phys. ReviewB 6, 1, p.110 (1972).

22. A.J.Dolgert, T.Blum, A.T.Dorsey, M.Fowler, Nucleation and growth of the superconducting phase in thepresence of a current,Phys. Review B57, 9, p.5432 (1998).

23. Q.Du, M.D.Gunzburger, J.S.Peterson, Analysis and approximation of the Ginzburg-Landau model for super-conductivity,SIAM Review, 34 No.1, pp.54 (1992).

24. T.P.Orlando, K.A.Delin,Foundations of Applied Superconductivity, (Addison-Wesley, Chap. 6-10, 1991).

25. T.P.Orlando, K.A.Delin,Foundations of Applied Superconductivity, (Addison-Wesley, Chap. 7.2, 1991).

26. R. Kato, Y. Enomoto, S. Maekawa, Effects of the surface boundary on the magnetization process in type-IIsuperconductors,Phys. Rev. B47 13, p.8016 (1991).

27. Q.Du, M.D.Gunzburger, J.S.Peterson, Computational simulation of type-II superconductivity including pin-ning phenomena,Phys. review B51 No.22, p.16194 (1995).

28. M.Mu, Y.Huang, An alternating Crank-Nicolson method for decoupling the Ginzburg-Landau equations,SIAM J. Numer Anal., 35, 5, p.1740 (1998).

29. F.Liu, M.Mondello, N.Goldenfeld, Kinetics of the Superconducting Film,Phys. Rev. Lett.66, 23, p.3071,(1991).


30. Y. Enomoto, R. Kato, Computer simulations of a type-II superconductor in a magnetic field,J. Phys. Condens.Matter 3, p.375 (1991).

31. M. Machida, H. Kaburaki, Direct Simulation of Time-Dependent Ginzburg-Landau Equation for Type-IISuperconducting Thin Film: Vortex Dynamics and V-I Characteristics,Phys. Rev. Lett.71, 19, p.3206, (1993).

32. L.Lapidus, J.H.Seinfeld,Numerical Solution of Ordinary Differential Equations, (Academic press, New York,1971).

33. K.Sepehrnoori,Runge-Kutta formulas with extended regions of stability, (TICAM report, 78-10, May 1978).

34. W. B. Richardson, G. F. Carey, and B. J. Mulvaney, Modeling Phosphorus Diffusion in Three Dimensions,IEEE Trans. on CAD.8, p.487 (1992).

35. G. F. Carey.Computational Grids: Generation, Adaptation, and Solution Strategies. Taylor and Francis,1997.

36. A. L. Pardhanani, G. F. Carey, A. Ardelea and W. B. Richardson, Modeling and Simulation of GinzburgLandau Models for Superconductivity,TICAM Report (in preparation), University of Texas, Austin, 2001.

37. C.Bolech, G.Buscaglia, A.Lopez, Numerical simulationof vortex arrays in thin superconducting films,Phys.Rev. B52, 22, p.15719 (1995).

38. G.F.Carey, W.B.Richardson, C.S.Reed, B.J.Mulvaney, Circuit, Device and Process Simulation, (John Wiley& Sons , 1996).

39. M. Machida, H. Kaburaki, Direct Numerical Experiment onTwo-Dimensional Pinning Dynamics of a Three-Dimensionnal Vortex Line in Layered Superconductors,Phys. Rev. Lett.75, 17, p.3178 (1995).

40. F. Bethuel, H. Brezis, F. Helein,Ginzburg-Landau Vortices, (Birkhauser, 1994).

41. M. Machida, H. Kaburaki, Structure of Flux Lines in Three-Dimensional Layered Type-II Superconductor:Numerical Experiments,Phys. Rev. Lett.74, 8, p. 1434 (1995).

42. J. W. Neuberger, R. J. Renka, Sobolev Gradients and the Ginzburg-Landau Functional,SIAM J. Sci. Comput.20, pp. 582 (1998).


TABLE 1Performance comparison of 1st and 4th order integration methods for representative

problem on domain of24� 24 scaled length units with different choices ofmesh resolution.

mesh order CPU time (sec.) # of steps % rejected steps64 � 64 4th order 2730.77 16548 30.791st order 411.51 2254 46.49200 � 200 4th order 20672.75 105592 22.851st order 5468.21 35096 17.49256 � 256 4th order 47892.98 168784 28.031st order 12167.23 51776 20.43

TABLE 2Comparison of Adams versus BDF integration schemes in 3D. NCF is the number of

nonlinear convergence failures. Adams is much more effecient for the giventolerances.

CPU Steps NCF20 � 20� 10 Adams 214 609 7mesh BDF 640 1023 28330 � 30� 15 Adams 860 1029 13mesh BDF 2367 1638 46940 � 40� 20 Adams 6050 2105 10mesh BDF 14435 2437 670

20 ! Please write\authorrunninghead{<Author Name(s)>} in file !

0 6 120

6

12

0 6 120

6

12

0 6 120

6

12

0 6 120

6

12

(a) (b) (c) (d)

FIG. 1. Contour plots ofj j from Eq.(5)-(7) at different times for square domain of side-length12 physicalunits with� = 2 andHe = 0:95 = 0:475Hc2. The mesh contains256 � 256 uniformly spaced nodes. Thetimes are(a) t = 20, (b) t = 80, (c) t = 350 and ,(d) t = 1000.

0 6 120

6

12

0 6 120

6

12

0 6 120

6

12

(a) (b) (c)

FIG. 2. Magnitude ofj j att = 500 for the model given by Eq.(8)-(9) with� = 2, �He = 0:44 = 0:22Hc2on a12 � 12 domain with three different uniform meshes: (a)32 � 32, (b) 64 � 64, and (c)128 � 128. Thisillustrates the nature of the coarse grid spurious solutionbehavior.

! Please write\titlerunninghead{<(Shortened) Article Title>} in file ! 21

0 6 120

6

12

0 6 120

6

12

(a) (b)

0 6 120

6

12

0 6 120

6

12

(c) (d)

FIG. 3. Magnitude ofj j from from Eq.(5)-(7) forHe = 0:95 = 0:475Hc2 on a12 � 12 domain. Themeshes densities are(a) 64� 64, (b) 90 � 90 (c) 128 � 128, (d) 256 � 256. The result in (a) is a snapshot att = 400 indicating clearly a non-physical solution. The depleted regions become wider (very slowly) until thetrivial solution j j = 0 is obtained everywhere. (b)-(d) represent long-term invariant configurations (no changeover several102 units of time).

0 6 120

6

12

0 6 120

6

12

0 6 120

6

12

0 6 120

6

12

(a) (b) (c) (d)

FIG. 4. Magnitude of for applied fieldHe = 0:80 = 0:40Hc2 on200� 200 mesh at the following timeinstants: (a) t=60, (b) t=70, (c) t=200, and (d) t=4000.

22 ! Please write\authorrunninghead{<Author Name(s)>} in file !

0 6 120

6

12

0 6 120

6

12

0 6 120

6

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0 6 120

6

12

(a) (b) (c) (d)

FIG. 5. j j with applied field increased toHe = 0:84 = 0:42Hc2 on 200 � 200 mesh at: (a) t=20, (b)t=30, (c) t=200, and (d) t=4000.

0 6 120

6

12

0 6 120

6

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0 6 120

6

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0 6 120

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(a) (b) (c) (d)

FIG. 6. j j for applied fieldHe = 0:90 = 0:45Hc2 on300� 300 mesh at: (a) t=20, (b) t=40, (c) t=400,and (d) t=4000.

0 6 120

6

12

0 6 120

6

12

0 6 120

6

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(a) (b) (c)

FIG. 7. Vortex configurations from Eq.(5)-(6) with� = 2, for three values of the external applied field:(a) He = 0:80 = 0:40Hc2, (b) He = 0:95 = 0:475Hc2 and (c) He = 1:10 = 0:55Hc2. The grids are128 � 128, 200 � 200 and304 � 304 respectively.

! Please write\titlerunninghead{<(Shortened) Article Title>} in file ! 23

0 3 60

3

6

0 3 60

3

6

(a) (b)

FIG. 8. Contour plot ofj j from the model Eq.(5)-(7) for� = 2,He = 0:88 = 0:44Hc2 on a12� � 12�domain at (a)t = 160 and (b)t = 400

FIG. 9. Vortex filaments in 3D with physical parameters� = 2, t = 100, and = 6:67 � 6:67 � 3:34in � units. In the left panel the external field isHe = (0:3; 0:3; 1) = (0:15; 0:15; 0:5)Hc2, while in the rightpanelHe = (0:6; 0:6; 1) = (0:3; 0:3; 0:5)Hc2.

numerical discretization and simulation of ginzburg landau...

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