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
