Adaptive Mesh Refinement for Multiscale Nonequilibrium Physics

Along-standing challenge in both ap-plied and theoretical physics is to fullyunderstand and predict the behaviorof systems far from thermodynamic

equilibrium,1–4 including those systems drivenby an external force or experiencing a suddenchange in environment (such as pressure or tem-perature). They also include systems transition-ing from one metastable or long-lived state toanother. The need to accurately model and nu-merically simulate these processes has becomemore pressing as efforts to derive well-parameterized evolution equations for meso-scopic nonequilibrium dynamical processes’progress. Simultaneously, it’s becoming increas-ingly important (for example, in phase transitionkinetics5,6) to be able to validate equations andtheir solutions against experiment. This valida-tion requires highly accurate and efficient par-tial differential equation (PDE) solvers.

PDE integrators for these problems aren’t typi-

cally selected for their space and time accuracy. In-stead, researchers have used easily implemented,mostly explicit methods to determine universal fea-tures such as domain growth exponents. For mod-ern materials applications, however, space and timeresolution are crucial for modeling system behav-ior. For these applications, the standard low-orderor fixed-grid integration approach is prohibitivedue to time-step constraints. We address this issueby applying adaptive mesh refinement (AMR)methods in dynamical condensed-matter systems.For ease of exposition, we focus on the time-de-pendent (real) Ginzburg-Landau (TDGL) equa-tions because they’re the prototypical example of aspatially extended nonequilibrium system under-going a dynamical phase transition.

Nonequilibrium PhenomenaThe equations of dynamical critical phenomena,phase separation, and so on typically fall into thereaction-diffusion or reaction-diffusion-advectiontype.4,7 They can contain several scalar, vector, ortensor fields, and they’re often dissipative, butsometimes they have an inertial manifold.8–10 Inthis article, we consider nonequilibrium phenom-ena of the following type:

• The system (for example, a binary alloy) is athigh temperature, in the disordered or singlephase.

24 COMPUTING IN SCIENCE & ENGINEERING

ADAPTIVE MESHREFINEMENT FOR MULTISCALENONEQUILIBRIUM PHYSICS

M U L T I P H Y S I C SM O D E L I N G

In this article, the authors demonstrate how to use adaptive mesh refinement (AMR) methodsfor the study of phase transition kinetics. In particular, they apply a block-structured AMRapproach to investigate phase ordering in the time-dependent Ginzburg-Landau equations.

DANIEL F. MARTIN AND PHILLIP COLELLA

Lawrence Berkeley National LaboratoryMARIAN ANGHEL AND FRANCIS J. ALEXANDER

Los Alamos National Laboratory

1521-9615/05/$20.00 © 2005 IEEE

Copublished by the IEEE CS and the AIP

MAY/JUNE 2005 25

• The temperature or other control parameter israpidly changed to a value at which the system isunstable to fluctuations.

• Via nucleation or continuous ordering, coherentstructures of the new stable phase grow.

• As time progresses, the ordered domains R grow,typically as some power of time R(t) ~ t.

Because most of the alloy concentration’s varia-tion is at the interface between domains, it’sworthwhile to use numerical and analytical meth-ods that take advantage of this fact. Relatively re-cent efforts work in the regime of a very sharp in-terface.11–13

One such approach is AMR, in which the com-putational mesh is locally refined in regions wheregreater accuracy is desired. Local refinement of thecomputational mesh can permit solutions with theaccuracy of the finest mesh with only a fraction ofthe cost of a computation on the equivalent uni-formly fine mesh. We use Marsha Berger andPhillip Colella’s block-structured approach,14 inwhich refinement is organized in logically rectan-gular regions of the domain. Discontinuities in thecomputational mesh due to local refinementrequire special stencils to maintain accuracy. Block-structured refinement allows this overhead at in-terfaces between coarse and fine meshes to beamortized over many regular operations on the in-teriors of the refined regions. This approach wasinitially implemented for gas dynamics,14 but hasbeen successfully extended to other systems ofequations including incompressible fluid dynam-ics,15,16 plasma fluid dynamics,17 and magnetohy-drodynamics (MHD).18

The Time-Dependent Ginzburg-Landau EquationsOur physical system is characterized by a coarse-grained local order parameter (x, t), whichrepresents a concentration difference betweencompeting phases.

The free energy for this system is given by

, (1)

where V() is a local potential. For simplicity, wechoose V() = (1/4)4 – (1/2)2, which is a double-well potential.

The standard approach to generating a dynam-ics when the order parameter isn’t conserved is toequate the local order parameter’s time derivativewith the local chemical potential, which in this case

is the (negative) of the free energy’s variational de-rivative with respect to the order parameter.Hence,

. (2)

For simplicity, we’ve set any kinetic coefficients tounity. Our specific potential V then yields the fol-lowing dynamics for the order parameter:

. (3)

These equations model the time development ofthe concentration difference between competingstable phases. Other evolution equations for non-equilibrium dynamics have similar behavior andwill be dealt with elsewhere.

AMR Operators and NotationAs Figure 1 illustrates, we start with a single, uni-form Cartesian mesh, with grid spacing h0 and in-dexing i = (0, N0 – 1). Where refinement is desired,we refine logically rectangular patches by a refine-ment ratio nref

0 , with cell spacing h1 = h0/nref0 . The

collection of logically rectangular patches with h =h1 are considered to make up a single AMR level,level 1 (the original base mesh is level 0). If furtherrefinement is desired, we can refine logically rec-tangular patches of level 1 by nref

1 to create a secondlevel of refinement (level 2).

∂∂

= ∆ − +φ φ φ φt

3

∂∂

= − = ∆ −φ δ φδφ

φ φt

FV

[ ]'( )

F dr V[ ]( )

( )φ φ φ= ∇ +

∫

2

2

x

y

Figure 1. Block-structured local refinement. Two levels of refinement(levels 1 and 2) are nested within the coarse mesh (level 0).


In general, we can create an arbitrary numberof refinement levels. The refinement level is re-lated to the coarser level ( – 1) by the refinementratio nref

–1 = h–1/h. Because the solution on finergrids is assumed to be more accurate, only cellsthat aren’t covered by refinement are consideredto have “valid” solution values; the collection ofcells on a level uncovered by refinement are con-sidered to be the “valid region” on a given AMRlevel. In our implementation, we’ve used refine-ment ratios that are powers of two. To simplify thestencils at the interfaces between fine and coarseregions, we require that refined regions be prop-erly nested14—the edge of level must either beadjacent to level ( – 1) cells or be located at theboundary of the computational domain. Imple-mentation is also greatly simplified by the use of

a global index space on each level; for the cell-centered discretization used in this work, the celli = (i, j) on level is centered at xi = (xi, yj) = ((i +1/2)h, (j + 1/2)h).

Implementing numerical algorithms on locallyrefined meshes requires care in the construction ofoperator discretizations at the coarse–fine interfaceto ensure that the appropriate accuracy is main-tained for the resulting solution. We refer to suchdiscretizations on the composite mesh hierarchy ascomposite or AMR operators. We’ll use a compositediscretization of the Laplacian operator,16 whichwe’ll refer to as Lcomp.

Although other AMR time-accurate applicationsemploy refinement in time and space,14–18 we’veelected to advance cells at all levels of refinementwith the same time step (also referred to as a non-subcycled algorithm). This approach has been used invarious contexts, including incompressible flow andporous media flow.19,20 The hyperbolic nature ofthe equations solved elsewhere14–16 make refine-ment in time particularly useful, because it enablesall levels to advance at similar CFL (Courant-Friedrichs-Lewy) numbers.

Because the advection schemes employed inthese other works have better phase-error accu-

racy at moderate CFL numbers, subcycling isimportant to the numerical accuracy on coarserlevels for these algorithms.1 Subcycling is partic-ularly useful when similar temporal accuracy is re-quired on all levels in the adaptive mesh hierarchyor when a problem has a very large range ofdynamic scales (resulting in many levels of refine-ment). The TDGL equations solved in this arti-cle produce solutions that quickly evolve into afairly sharp interface between regions in whichthe solution is relatively constant spatially. In thiscase, the solution’s accuracy on the coarse levelsisn’t important because all the dynamics occur atthe interface. Therefore, implementing subcy-cling won’t affect solution accuracy in any appre-ciable way, assuming that the interface betweenthe two phases is completely refined. Subcycledalgorithms are more complex, both algorithmi-cally and in code implementation. In the case ofthe TDGL equations, the subcycled algorithm fora given level () would look like this:

1. If ( > 0), interpolate boundary conditions intime and space from level ( – 1).

2. Advance the solution on level using a single-level advance from time (t) to time (t + t),where t = (1/nref

–1)t–1.3. If a level ( + 1) exists, advance the solution on

the next finer level nref times (which will also

include advancing any levels finer than ( + 1)).At this point, both the level and level ( + 1)solutions will have reached the same time.

4. Perform a multilevel elliptic synchronizationsolve over all levels that have reached time (t

+ t).5. Average level ( + 1) solution to covered re-

gions of level .

Note that in a subcycled algorithm, the TDGLequations’ elliptic/parabolic nature implies that thesynchronization step entails an extra multilevel el-liptic solve to enforce the proper matching condi-tions between coarse and fine levels.15 In contrast,the nonsubcycled algorithm is the same as thesingle-level algorithm, except that it uses compos-ite AMR operators.

Because there is no real accuracy benefit to im-plementing subcycling for the solutions computedin this work, we decided to implement the simplerchoice. Adopting a nonsubcycled algorithm resultsin a much simpler algorithm with fewer ellipticsolves; we believe that the benefits of refinement intime are outweighed by the increased algorithmiccomplexity and computational cost the additionalelliptic solves would require in this case.

Subcycling is particularly useful when temporal

accuracy is required on all levels in the adaptive

mesh hierarchy or when a problem has a very

large range of dynamic scales.

MAY/JUNE 2005 27

IntegrationWe wish to compute solutions to the TDGLequations that are second-order accurate in timeand space. Due to the stability constraints of ex-plicit methods for parabolic equations, we electto use a semi-implicit approach instead. Althoughthe Crank-Nicolson scheme is commonly usedfor these types of problems, it suffers from accu-racy and stability problems in the presence of lo-cal refinement. Instead, we use the second-orderL0-stable scheme detailed elsewhere.21 Althoughthis scheme requires two elliptic solves for eachupdate instead of the one required for Crank-Nicolson, its improved performance in the pres-ence of local refinement makes it a good choicefor AMR algorithms.

Mathematically, we can treat the TDGL equa-tions as a heat equation with a nonlinear forcingterm:

, (4)

where V() is the potential.We discretize in time and space in a cell-

centered manner:

ni = (xi, tn). (5)

The basic advance step from time tn to time tn+1

= tn + tn proceeds as follows:

1. Evaluate source term Sn = –(V)n = –dV/d(n).2. Predict ~

n+1 using a forward-difference ap-proximation:

~n+1 n + tn( – (V)n). (6)

3. Use ~(n+1) to predict source term Sn+1 = –(V )n+1

for time tn+1.4. Do initial elliptic solve for parabolic advance

and solve for intermediate term e:

(I – r2tLcomp)e = e (7)

e = (1 + (1 – a)tLcomp)n

+ 1/2t [Sn + (1 – 2(a – 1/2)tLcomp)Sn+1]. (8)

5. Do second elliptic solve for parabolic advanceand solve for n+1:

(I – r1tLcomp)n+1 = e. (9)

The quantities a, r1, and r2 are the values suggestedelsewhere:21

a = 2 – – ,

discr = ,

r1 = ,

r2 = ,

where is a small quantity (we use 10–8).

SolverWe constructed our implementation of this algo-rithm with the Chombo infrastructure,22 whichsimplifies the implementation of the locally adap-tive algorithm. To perform the multilevel AMR el-liptic solves, we use the Chombo AMR ellipticsolver, which is based on a multigrid algorithm.

ResultsFor this AMR approach to be worthwhile, it mustsatisfy three things:

1. The AMR algorithm should converge at sec-ond-order rates; adding coarse–fine interfacesto the discretization can cause a loss of accu-racy, which can impact convergence.

2. Suitably placed local refinement should resultin the same solution accuracy as the solutionon an equivalent uniformly fine computationmesh.

3. The use of local refinement should result incomputational savings over the uniform meshsolution with the same resolution.

Let’s examine each of these in more detail.

Convergence of the AMR Algorithm To demonstrate this algorithm’s convergenceproperties, we solve the sample three-dimen-sional TDGL system (Equation 3) on a 64.0 64.0 64.0 domain using an initial condition inwhich is initialized to be a sum of sinusoids tocreate an analytic initial condition that is alsocomplex geometrically.

To evaluate convergence consistently, we specifya refined patch in the center of the domain and re-fine the entire grid hierarchy. Because the gridhierarchy in this case isn’t truly adaptive, we don’texpect to get fine-grid accuracy in these computa-tions (although demonstrating the algorithm’soverall convergence properties is useful). The so-

2 1aa discr

−−

2 1aa discr

−+

a a2 4 2− +

2

∂∂

= ∆ −φ φ φφt

dVd

( )

lution features cross coarse–fine interfaces in thistest, so this test highlights any discretization prob-lems at the coarse–fine interfaces. Because no ana-lytic solution exists for this problem, we computea solution on a uniform 5123 fine mesh and treat itas the “exact” solution against which we compareother computed solutions. We compute errornorms by averaging the “exact” solution down tothe appropriate resolution, subtracting the com-puted solution, multiplying by the cell volume oneach level, and then summing over the valid re-gions on each level:

. (10)

Because the domain is a 64 64 64 cube, the L1

and L2 error norms are normalized by dividing bythe total domain volume, V = 262,144.

Tables 1, 2, and 3 show the test problem’s con-vergence at L1, L2, and L; note that because therefinement is placed without regard to the solu-tion, there is no apparent accuracy gain due to re-finement. Because the time step and cell spacingare reduced simultaneously (the time step ishalved when the cell spacing is halved), second-order convergence in these plots demonstratesconvergence in both time and space. As expected,adding refinement in this example doesn’t no-ticeably improve the solution’s accuracy, becausethe refinement isn’t placed in an adaptive manner.

AMR AccuracyFor the AMR approach to be worthwhile, a solu-tion with well-placed local refinement should pro-vide accuracy essentially equivalent to a solutioncomputed with a uniform fine mesh. To demon-strate this, we use a different test problem with amore AMR-appropriate initial condition.

LV

h AvgAMR exactvalid2

3 2

0

1φ φ φ= −=

| ( ) | on Ω

mmax

∑

12

LV

h AvgAMR exactvalid1

3

0

1φ φ φ= −=

| ( ) |

m

on Ωaax

∑


Table 1. Convergence results for L1 arbitrary refine grids, L1 norm.

Base grid size Single level nref = 2 nref = 4Error Rate Error Rate Error Rate

32 1.0053e-01 — 9.0256e-02 — 8.7002e-02 —64 1.6024e-02 2.65 1.6800e-02 2.42 1.5859e-02 2.46128 3.9268e-03 2.02 3.8158e-03 2.13 — —256 9.9205e-04 1.98 — — — —

Table 2. Convergence results for arbitrary refined grids, L2 norm.


32 3.0811e-04 — 3.1077e-04 — 3.0196e-04 —64 5.4665e-05 2.49 5.7526e-05 2.43 5.4348e-05 2.47128 1.3575e-05 2.01 1.3121e-05 2.12 — —256 3.4496e-06 1.98 — — — —


32 1.0257e+00 — 1.0840e+00 — 1.0617e+00 —64 3.0519e-01 1.75 3.3780e-01 1.68 3.2419e-01 1.71128 7.8087e-02 1.97 7.8628e-02 2.10 — —256 1.9702e-02 1.99 — — — —

Table 3. Convergence results for arbitrary refined grids, L norm.

MAY/JUNE 2005 29

The initial condition for is the sum of twoGaussians in a 64.0 64.0 64.0 cubic domain:

, (11)

where

n = 1x0 = (30, 24, 24)0 = 10 = 25x1 = (36, 40, 40)1 = –11 = 25.

The solution is evolved to time t = 1, using theundivided gradient of as the indicator for refine-ment. Tables 4, 5, and 6 show the error normscomputed by using Equation 10; base grid size isthe number of cells per side on the coarsest AMRlevel. If the refinement is effective, then the 323

base grid solution with nref = 2 should have thesame error as the 643 uniform grid solution, andthe 323 with nref = 4 solution should have the sameaccuracy as the 1283 uniform grid solution. Whenthe refinement is appropriately placed, Tables 4, 5,

and 6 demonstrate that the solutions do attain theequivalent uniform grid solution’s level of accuracy.

PerformanceTo demonstrate the AMR approach’s effective-ness, we use a simple test problem on a 64 64 64 domain with homogeneous Dirichlet bound-ary conditions.

n = 1x0 = (30, 24, 24)0 = 10 = 50x1 = (36, 40, 40)1 = –1 1 = 50.

We compute this with a base 323 mesh (h0 = 2.0),with two levels of refinement at nref = 4. As before,we tag cells for refinement based on the undividedgradient of . To estimate the cost of the equiva-lent uniform-grid solution, we computed 15 timesteps on the equivalent 512 512 512 uniformgrid. Figure 2 shows the solution at the initialtime and at time t = 10.0. Figure 3 shows thefinest-level grid boxes at the final time (by thispoint, the first level of refinement has grown tocover the entire domain).

φ σ ρ( , )0

2

0x

x x

=

−

=∑ ii

ne

i

i

Base grid size Uniform grid error nref = 2 error nref = 4 error32 7.0774e-04 2.2495e-04 6.3469e-0564 2.2375e-04 6.1733e-05 1.5261e-05128 5.3856e-05 1.3769e-05 —256 1.3174e-05 — —

Table 4. Errors for Gaussian AMR test (L1).


Table 5. Errors for Gaussian AMR test (L2).


Table 6. Errors for Gaussian AMR test (L).


Figure 4 shows the total number of cells ad-vanced per time step for the AMR computationcompared to the uniform grid computation. Thisnumber slowly increases to approximately 40 per-cent of the number in the uniform grid case (atthe initial time, it’s less than 1 percent of thenumber of cells in the uniform grid case). Thesmaller problem size can be crucial in fitting theproblem onto machines with less memory. Thetotal time spent advancing the solution (notcounting I/O and initialization) was 87,942.67seconds on a 1.8-GHz Opteron machine. As acomparison, running the equivalent single-levelcomputation out to time t = 10 would have taken291,360 seconds. The AMR computation took30.2 percent of the execution time and required

only 40 percent of the memory the equivalentsingle-level computation needed. We shouldnote, though, that this is still a fairly small sam-ple problem. We expect that a larger problemwith finer resolution will result in more pro-nounced benefits due to AMR.

The work described here onlyscratches the surface of potentialAMR applications to nonequilib-rium problems. A first step beyond

would be to treat systems with multiple-orderparameter fields, including, for example, the dy-namics of liquid crystals and superfluids.

Another direction in which AMR could be ap-plied is modeling systems with a conserved orderparameter. The dynamical equations then containhigher order differential operators such as theCahn-Hilliard equation, in which the Laplacianis replaced by the fourth-order (–) operator.Classical finite difference spatial discretizationsaren’t well-suited for this problem. Instead, wewould consider using a different approach tomultiresolution modeling for constant coefficientPDEs.23

For the problem examined in this work, the undi-vided gradient of the solution provided a reasonableindicator for locating refinement. For more compli-cated problems, this simple approach could be re-placed by more appropriate indicators. One ap-proach14 is to estimate the local truncation error andrefine it where it is above a given threshold. Anotherapproach is to determine a physical parameter of thesolution as an indicator for refinement (for example,vorticity in a fluid or heat release in a flame).

Acknowledgments This work, LA-UR-05-0749, was carried out under theauspices of the US Department of Energy at Los AlamosNational Laboratory and Lawrence Berkeley NationalLaboratory. Additional funding came from lab-directedresearch and development. It was supported at LBNL bythe US Department of Energy, Director, Office of Science,Office of Advanced Scientific Computing, MathematicalInformation, and Computing Sciences Division undercontract DE-AC03-76SF00098.

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(a) (b)

Figure 2. Gaussian test problem. The solutions for isosurfaces of =(–0.7, 0.7) at (a) the initial condition and (b) time t = 10.

Figure 3. Grid boxes on finest level at solution time t= 10. Grid boxes make up the finest level ofrefinement at solution time t = 10 for the two-Gaussian problem.

MAY/JUNE 2005 31

4. S. Dattagupta and S. Puri, Dissipative Phenomena in CondensedMatter: Some Applications, Springer-Verlag, 2004.

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7. C. Hohenberg and B.I. Halperin, “Theory of Dynamic CriticalPhenomena,” Rev. Modern Physics, vol. 49, no. 3, 1977, pp.435–479.

8. P. Constantin et al., Integral Manifolds and Inertial Manifoldsfor Dissipative Partial Differential Equations, Springer-Verlag,1989.

9. J.C. Robinson, Infinite-Dimensional Dynamical Systems: An Intro-duction to Dissipative Parabolic PDEs and the Theory of Global At-tractors, Cambridge Univ. Press, 2001.

10. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanicsand Physics, Springer-Verlag, 1997.

11. R.F. Almgren and A.S. Almgren, “Phase Field Instabilities andAdaptive Mesh Refinement,” Proc. Materials Week ’96: Mathe-matics of Microstructure Evolution, Mineral Metals and MaterialsSoc., 1996, pp. 205–214.

12. N. Provatas, N. Goldenfeld, and J.A. Dantzig, “Adaptive MeshRefinement Computation of Solidification Microstructures UsingDynamic Data Structures,” J. Computational Physics, vol. 148, no.1, 1999, pp. 265–290.

13. N. Goldenfeld, J.H. Jeong, and J.A. Dantzig, “Phase Field Modelfor Three-Dimensional Dendritic Growth with Fluid Flow,” Phys-ical Rev. E, vol. 64, no. 4, 2004, pp. 1–14.

14. M.J. Berger and P. Colella, “Local Adaptive Mesh Refinement forShock Hydrodynamics,” J. Computational Physics, vol. 82, no. 1,1989, pp. 64–84.

15. A.S. Almgren et al., “A Conservative Adaptive ProjectionMethod for the Variable Density Incompressible Navier-StokesEquations,” J. Computational Physics, vol. 142, no. 1, 1998, pp.1–46.

16. D. Martin and P. Colella, “A Cell-Centered Adaptive ProjectionMethod for the Incompressible Euler Equations,” J. ComputationalPhysics, vol. 163, no. 2, 2000, pp. 271–312.

17. P. Colella, M. Dorr, and D. Wake, “Numerical Solution of Plasma-Fluid Equations Using Locally Refined Grids,” J. ComputationalPhysics, vol. 152, no. 2, 1999, pp. 550–583.

18. R. Samtaney et al., “3D Adaptive Mesh Refinement Simulationsof Pellet Injection in Tokamaks,” Computers Physics Comm., vol.164, nos. 1–3, 2004, pp. 220–228.

19. R.D. Hornung and J.A. Trangenstein, “Adaptive Mesh Refinementand Multilevel Iteration for Flow in Porous Media,” J. Computa-tional Physics, vol. 136, no. 2, 1997, pp. 522–545.

20. M.L. Minion, “A Projection Method for Locally Refined Grids,” J.Computational Physics, vol. 127, no. 1, 1996, pp. 158–178.

21. E.H. Twizell, A.B. Gumel, and M.A. Arigu, “Second-Order, L0-Sta-ble Methods for the Heat Equation with Time-Dependent Bound-ary Conditions,” Advances in Computational Mathematics, vol. 6,nos. 3 and 4, 1996, pp. 333–352.

22. Applied Numerical Algorithms Group, The Chombo Frameworkfor Block-Structured Adaptive Mesh Refinement, tech. report,Lawrence Berkeley Nat’l Lab., 2005; http://seesar.lbl.gov/ANAG/chombo.

23. G.T. Balls and P. Colella, “A Finite Difference Domain Decom-position Method Using Local Corrections for the Solution of Pois-son’s Equation,” J. Computational Physics, vol. 180, no. 1, 2002,pp. 25–53.

Daniel F. Martin is a scientist in the Applied Numeri-cal Algorithms Group at the Lawrence Berkeley Na-tional Laboratory (LBNL). His technical interests include

applying adaptive mesh refinement techniques to thesolution of differential equations and implementinguseful scientific software. Martin has a PhD in me-chanical engineering from the University of California,Berkeley. Contact him at [email protected].

Phillip Colella is a senior staff scientist and groupleader of the Applied Numerical Algorithms Group atLBNL. His technical interests include numerical meth-ods for partial differential equations, high-resolutionfinite difference methods, adaptive mesh refinement,volume-of-fluid methods for irregular boundaries,and programming language and library design forparallel scientific computing. Colella has a PhD inapplied mathematics from the University of Califor-nia, Berkeley. He is a member of the US NationalAcademy of Sciences. Contact him at [email protected].

Marian Anghel is a technical staff member in the Com-puter and Computational Sciences Division at Los AlamosNational Laboratory. His technical interests include theapplication of statistical learning theory and numerical ac-celeration algorithms to the data-driven modeling andidentification of nonlinear, multiscale systems. He has aPhD in physics from the University of Colorado, Boulder.Contact him at [email protected].

Francis J. Alexander is a deputy group leader at LosAlamos National Laboratory. His technical interests in-clude statistical mechanics, computational physics, andestimation theory. Alexander has a PhD in physics fromRutgers University. Contact him at [email protected].

2e + 07

4e + 07

6e + 07

8e + 07

1e + 08

1.2e + 08

1.4e + 08

Num

ber

of c

ells

Solution time

Cells updated per time step

Adaptive mesh refinementUniform grid

0 4 6 8 102

Figure 4. Cells updated per time step. “Solution time” is the time t towhich the solution (x, t) has been evolved. “Uniform grid” is thenumber of cells in the equivalent uniform-grid computation.

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Adaptive Mesh Refinement for Multiscale Nonequilibrium Physics