Motivation and Overview Identifying Accuracy Bottlenecks ...€¦ · Development and Analysis from a Stochastic PDE Perspective Team-building Tutorials Analysis and Development Using

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: Hui.Wan@pnnl.gov, woodward6@llnl.gov

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