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The team: Hui Wan1 (PI), Carol Woodward2 (Co-lead), Jack Reeves Eyre3, David Gardner2, Huan Lei1, Vince Larson4, Phil Rasch1, Lance Rayborn1, Balwinder Singh1, Jeremy Sousa3, Panos Stinis1, Nicolas Strike4, Chris Vogl2, Xubin Zeng3, and Shixuan Zhang1
Affiliations: 1Pacific Northwest National Laboratory, 2Lawrence Livermore National Laboratory, 3University of Arizona, 4University of Wisconsin—Milwaukee. Contact: [email protected], [email protected]
Development and Analysis from a Stochastic PDE Perspective
Team-building Tutorials Analysis and Development Using Process-level Understanding
Integration into E3SMAccuracy Metrics and Testing Methods
Development and Analysis from a Deterministic PDE Perspective
Identifying Accuracy BottlenecksMotivation and Overview
Goal:Improvenumericalconvergencethroughidentificationofmathematicalsourcesofconvergencedegradationandimprovedmodelsofsub-gridprocesses
Approach:Conductanerroranalysisoftheintegrationschemeappliedinthesimplecondensationmodelandidentifyassumptionsforconvergencethatmaynothold.Basedonfindings,rederivethemodelparameterizationusingexplicitsub-gridprofiles,reconstructedfromgrid-cellaveragedvalues.
KeyResults:• Completederroranalysisofadditivelysplittwo-process
integrationschemewith/withoutsequentialsplittingandfinitedifference(FD)approximations
• Singularitiesandlimitedcontinuityinmodelterms,inparticularthecloudfraction,f,canreduceconvergenceorder
• DerivedandimplementedanewversionofthesimplecondensationmodelinE3SMtestedonCori
Firstresultswithrevisedmodelshowgoodconvergencebutlargeerrors;1-partitionprofilemakesassumptiononmodelstatebut2-partitiondoesnot
Figure5:Timesteppingerrorandnumericalconvergencerateoftheadvection-diffusionmodelwithandwithouttheItocorrectionterm.
Goal:Improvenumericalconvergencebyaccuratestochasticmodelingofsub-gridprocesses
Approach:Introduceastochastic(Ito)correctiontermtorepresenttheimpactoffastforcingonslowercomponentsofthefluid motion.
KeyResults:• Configuredanadvection-diffusionmodelwithawidespectrum
ofstate-dependentfastforcingasatestproblem• DemonstratedtheuseofItocorrectiontorestoreconvergence
forwhiteforcingspectra• GeneralizedItocorrectionforredforcingspectra;improved
convergenceandaccuracy(Figure5)
@u
@t
= �c
@u
@x
+ µ
@
2u
@x
2+ g(u)n(t)
n(t)- solutionu(x, t) - fastphysicalprocess
Advection-diffusionmodel:
combinedsplittingandFDerrortruncationerror FDerrorinitialerror 1st order
Goal:Isolatephysicalprocessesandcodepieceswithpoorconvergence.Providedetaileddescriptionofmodelequationsandcodeimplementationformathematicalandnumericalanalysis
Approach:Conductconvergencetests.Usephysicalinsightstoidentifypathologicalbehaviorandsimplifymodelequationsorcodetofacilitatefurtherinvestigations
KeyResults:• Identifiedandfixedasignificantbuginthesingle-columnmodel.
Demonstratedself-convergencetestasausefulwaytodetectissueshardtonoticeinthedefaultmodelconfiguration(Figure2)
• Constructedasimplifiedlarge-scalecondensationschemeandidentifieddependenceofconvergenceonmodelformulation,physics-dynamicscoupling,andtimesteppingwithintheparameterization.Demonstratedthatrestoringconvergencecanleadtosubstantialchangesinmodel’slong-termclimate(Figure3)
Figure3:Multi-yearzonalmeantotalcloudfractioninCAM4usingarevisedphysics-dynamicscoupling(red)thathelpedtorestore1st-orderconvergenceinasimplelarge-scalecondensationmodel.
Figure 6: 20-year mean cloud fraction change caused by reduction of model step size from 30 min to 5 min in E3SMv1.
Background:TheE3SMatmospheremodelcomprisesafluiddynamicssolver(thedynamicalcore)andtherepresentationofmanysub-grid-scaleprocesses(theparameterizations).Thelattertypicallyuses simple timeintegration methods and long step sizes.
Thechallenge:InE3SMv1andseveralofitspredecessors,time-stepconvergenceintheparameterizationsissignificantlyslowerthantheexpected1st-orderrate,andthetimesteppingerrorsaresubstantiallylargerthanthoseinthedynamicalcore(Figure1).Thisissueis
Projectobjectives:• Understandcausesofpoorconvergence• ImprovesolutionaccuracyinE3SM
also relevanttoprobablyallglobalandregionalatmosphericmodels.
Figure 1: Time-stepping errors and self-convergence rates in E3SMv1 atmosphere model.
Goal:OvercomethebarriersbetweentwodisciplinesApproach:Useinformalteamtutorialstoclarifylanguage,introducebasicconceptsandmethods.Topicscoveredtodate:• E3SMcodestructure• Timesteppingproblemsinatmosphericmodels• CLUBBandthefilteringapproach• Introductiontostabilityandconvergence• Introductiontostochasticdifferentialequations
andmodelreduction
Keyresults:• Quantifiedtime-stepsensitivityin
E3SMv1simulations(Figure6)• Implementedaclosureindeep
convectionparameterizationwithimprovedprecipitationoverthetropicalwestPacific
• Identifiedprocesscouplingissuesrelatedtoaerosolsandwatercycle
Goal:Establishadditionalmetricsformeasuringsolutionaccuracy
Keyresults:• Foundthemagnitudeof
clippingtermstobeagoodindicatorofnumericalproblem
• Foundsignificantcorrelationbetweenpoorconvergence,largetimesteppingerror,andfastgrowthofroundingerror
Figure 6: Initial results suggest that fast growth of rounding error can be an indicator of low accuracy in time stepping.
Goal:EnsureprojectfindingsareincorporatedintoE3SMKeyresults:• ProvidedmultiplebugfixestoE3SM• Currentlyincorporatingarecentversion
ofCLUBBintoE3SMasaGit subtree• Regularlyrebasingproject’scode
branchestotheE3SMmaster• Enablingall(includingmath)team
memberstomodifyandrunE3SM
FigurecourtesyofWikimediaCommons
11tutorialswitharchivedslidesandrecordings
Figure4:Firstselfconvergenceresultswithrevisedsimplecondensationmodel.
Withsufficientlysmalltimestepsizes,bothschemesexhibitfirst-orderconvergence.Whenthefastforcingcannotbefullyresolved,theItocorrectiontermimprovesthenumericalconvergenceratefromO(∆t0.2)toO(∆t0.5).
Strategy:• Closecollaborationamongatmosphericscientists,applied
mathematicians,andcomputationalscientists• Effortsareorganizedastasksdescribedintherestofthisposter• Newmodelformulationsandtimeintegrationmethodsarefirst
developedinsimplifiedmodelconfigurationsandthenimplementedandevaluatedinE3SM
Figure2:Timesteppingerrorandself-convergencerateinsingle-columnsimulationsofstratocumuluscloudsbeforeandafterabugfix.
OriginalRevisedBeforebugfix
Afterbugfix
