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Page 1: Stochastic Simulation of Reaction-Diffusion Systems: A ... · PDF fileAdd reactions ØInstead of using the chemical Langevin equation, which can give physically wrong results ... bell2_poster_ascr_ampi2017

DOEASCRAppliedMathematicsPrincipalInvestigators'(PI)Meeting,Rockville,MD,September11-12,2017

Approach

Abstract

Conclusions and Future Work

Motivation

Stochastic Simulation of Reaction-Diffusion Systems:A Fluctuating Hydrodynamics Approach

J. B. Bell1, C. Kim1, A. Nonaka1, A. L. Garcia2, and A. Donev3

1Lawrence Berkeley National LaboratoryCenter for Computational Sciences and Engineering

2San Jose State University, 3Courant Institute

Results

Areas in which we can help

References

Areas in which we need help

Wedevelopnumericalmethodsforstochasticreaction-diffusionsystemsbasedonapproachesusedforfluctuatinghydrodynamics(FHD).Ourformulationissimilartothereaction-diffusionmasterequation(RDME)descriptionwhenthestochasticPDEsarespatiallydiscretizedandreactionsaremodeledasasourcetermhavingPoissonfluctuations.However,unliketheRDME,whichbecomesprohibitivelyexpensiveforaincreasingnumberofmolecules,ourFHD-baseddescriptionnaturallyextendsfromtheregimewherefluctuationsarestrong,i.e.,eachmesoscopiccellhasfewreactivemolecules,toregimeswithmoderateorweakfluctuations,andultimatelytothedeterministiclimit.Inaddition,ourapproachcanbereadilygeneralizedtomorerealisticdiffusionmodelsandnaturallycoupledwith(fluctuating)hydrodynamicequations.

• C. Kim, A. Nonaka, J. B. Bell, A. L. Garcia, and A. Donev, “Stochastic simulation ofreaction-diffusion systems: A fluctuating-hydrodynamics approach”, J. Chem. Phys.146, 124110 (2017).

• A. K. Bhattacharjee, K. Balakrishnan, A. L. Garcia, J. B. Bell, and A. Donev, “Fluctuatinghydrodynamics of multi-species reactive mixtures”, J. Chem. Phys. 142, 224107 (2015).

• A. Donev, A. Nonaka, A. K. Bhattacharjee, A. L. Garcia, and J. B. Bell, “Low Machnumber fluctuating hydrodynamics of multispecies liquid mixtures”, Phys. Fluids 27,037103 (2015).

Modelingchemicalreactionsatdifferentscales• MacroscopicscalesØ Lawofmassaction→ODEsØ EfficientODEintegrationalgorithms

• MicroscopicscalesØ ChemicalmasterequationØ ModelasMarkovprocess(SSA)

• Atintermediatescales,lawofmassactionisnotcorrectbutSSAistooexpensive.

• NewapproachbasedonfluctuatinghydrodynamicsØ StochasticPDEfordiffusionØ Tau-leapingtreatmentofchemistryØ Generalizedtoincludeeffectsoffluidflow

SpatialDiscretization(setofstochasticODEs)Cartesiangridfinitevolumeapproachfornumberdensities1.Diffusion-onlySPDEs

Ø ValiditybeyondtheGaussianapproximation,includingnon-negativity,dependsontheformofthefaceaverage,,usedinthestochasticmassflux.

2.Addreactions

Ø InsteadofusingthechemicalLangevinequation,whichcangivephysicallywrongresults,weusethetau-leapingmethod.

2DPatternformationWetestournumericalmethodsonatime-dependentproblemandcomparethemwithareferenceRDME-basedmethod.

1DSchlöglmodelWefirstvalidateourformulationandnumericalschemesusinganalyticresultsforsteady-statepropertiesofthissimplemodel.

3DfrontpropagationWedemonstratethescalabilitytolargesystemsandcomputationalefficiencyofourFHDapproachusingthis3-dimensionalexample.Thesystemisdividedinto2563 cellsandinvolvestheequivalentof1012 molecules.

Chemo-hydrodynamicinstabilities(ongoingwork)Wegeneralizeourformulationsothatreactionanddiffusionarecoupledwithfluidflow.OneinterestingexampleistheformationofasymmetricconvectivefingersobservedintheRayleigh-Taylorinstabilitywhencoupledtoaneutralizationreaction.

Wecombinetheefficiency ofthefluctuatinghydrodynamicsapproach(→diffusion)andtherigor ofthemasterequationapproach(→reaction).

Temporalintegrator(implicitmidpoint+tau-leapingscheme)

Ø Deterministicdiffusionpartistreatedimplicitly→stabilitylimitisbypassed.Ø Second-orderweakaccuracyforlinearizedequationsØ WeimplementthealgorithmusingtheAMReXsoftwareframework.

(availableathttps://github.com/BoxLib-Codes/FHD_ReactDiff.git)

1D

WecharacterizethepatternformationusingthedecaypatternoftheaverageconcentrationofUmolecules.

ThisstatisticalanalysisshowsthatourFHD-basedmethodreproducestheRDMEresults.However,ourmethodismuchfaster.Iftherearemanymoleculespercell(→largecross-sectionvalueA),theRDME-basedalgorithmistooslow.

Thermodynamicequilibrium,strongfluctuations:For10moleculespercell,ourmethodaccuratelyreproducesthePoissondistribution.Thisisnottrivialatallbecauseweusecontinuous-rangenumberdensityandGaussianwhitenoise.Thisalsovalidatesourchoiceforthefaceaverage,.

Outofequilibrium,weakfluctuations:Linearizedanalysisonthestructurefactorshowsthatourimplicitscheme(ImMidTau)givesveryaccuratestructurefactorseveniftimestepsizeismuchlargerthanthestabilitylimit∆tmax.Theresultingstructurefactoristhird-orderaccurate.

ImMidTau∆t = 5 ∆tmax

ExMidTau∆t = 0.5 ∆tmax

Stochasticreaction-diffusion

Correspondingdeterministicsimulationwiththesamenoisyinitialcondition

t = 8 s t = 16 s

NaOH (aq) (denser)

HCl (aq)

HCl + NaOH → NaCl + H2O Corresponding no-reaction case

Basedonouranalyticalandnumericalinvestigation,weconcludethatouralgorithm(ImMidTau)isanefficientandrobustalternativenumericalmethodforstochasticreaction-diffusionsimulations.

1. Computationalcostdoesnotincreaseforincreasingnumberofmoleculespercell.Hence,ourmethodcanefficientlysimulatereaction-diffusionsystemsoverabroadrangeofrelativemagnitudeofthefluctuations.

2. TheschemeallowsasignificantlylargertimestepsizewithoutdegradingaccuracycomparedtoexistingRDME-basednumericalmethods.

3. OurFHD-basedapproachcantakeadvantageofefficientparallelalgorithms.

4. Themethodhasbeengeneralizedtomorecomplexproblemsthatincludefluidflowwithrealisticmulti-componentdiffusion.

Futurework:ionicspecieswillbeincludedforthesimulationofelectro-chemicalphenomenainelectrolytesolutions.

• Mesoscopicmodelingofreactions• Stochasticsimulationalgorithms• Hybridalgorithms

• Improvedlinearsolversforevolvingarchitectures• ProgrammingmodelstosupportGPU

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