Multi-adaptive Time-Stepping - FEniCS Project

Computability of the Lorenzsystem and the Solar systemAnders Logglogg@math.chalmers.seChalmers Finite Element CenterAnders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.1/37

¡¢ £¤OutlineMulti-adaptive GalerkinHigher-order elementsSolving the Lorenz system onA few words about the Solar systemAnders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.2/37

£'& (¦!#"Ordinary Galerkin:for©¦¨§¥Ordinary Galerkin© £ ©¦ £ ¦ ©£ ¦and the trial and testwith ,spaces defined as©¦¢©$£¦!#"¤© ¦¡¢ £ ©$)% Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.3/37

£+& (¦! "Multi-Adaptive Galerkin:©¦¨§¥ *Multi-adaptive Galerkin© £ ©¦ £ ¦ ©£ ¦with , and the trial and testspaces now defined as©¦¢©$£¦! "+¤© +¦¡¢ £ ©$)%+ Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.4/37

! ",¢Individual Timestepslacements- "Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.5/37

1.05.2© 7The Discrete Equations,on! "Making an ansatz for©¦© £2 + 3 .¦4" 0 "+./©¦we end up with £©©¦© ¦ £2 + 3 .¦4"0 "+.(¦¢ £&63 . $ 5 for certain weight functionsAnders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.6/37

0 "9" 0& %+5*© ¦¨§8 523 . $6 £© ¦¢ £7 ©The Discontinuous MethodSimilarly for the multi-adaptive version of thediscontinuous method, , we obtain© ¦© ¦ £2 + 3 9¦4"for certain weight functions.Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.7/37

Multi-adaptive Time-SteppingAction: Redoing stepst = 0n = 0lastAnders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.8/37

Multi-adaptive Time-SteppingAction: Redoing stepst = 0.01n = 1Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.8/37












Multi-adaptive Time-SteppingAction: Redoing stepst = 0.01n = 2.6667Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.8/37











Multi-adaptive Time-SteppingAction: Clearing stepst = 0.01n = 2Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.8/37



Multi-adaptive Time-SteppingAction: Coveringt = 0.02n = 2Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.8/37














































Multi-adaptive Time-SteppingAction: Clearing stepst = 0.02n = 3.3333Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.8/37




Multi-adaptive Time-SteppingAction: Coveringt = 0.06n = 3.3333Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.8/37

































































































































Multi-adaptive Time-SteppingDonet = 0.1n = 2firstAnders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.8/37

:§© ¦¨§¥ *:§7© ¦¨§8*7;¢¥ *General OrderTheof order:§"method is of order., i.e. locallyThe method is of orderlocally of order .:§", i.e.Simple Galerkin thus gives methods of arbitraryorder with individual timesteps, such as e.g. the:th order method .© ¦:¢Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.9/37

C C C C@ ¦ £DFEGIHD C C C C©/(J * KL- + + $ Galerkin ErrorsMany different variantsMore or less local/globalTwo extremes (for the continuous method):Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.11/37

MA O ("&© E¦Computational ErrorsA £ * KL 32NJPO 32/(where¦ £ ©¦ "© EA 7 - "Q+is the discrete residual. (For the discontinuousmethod, ("&are replaced by("&.)& "£ & O "£ OAnders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.12/37

2NJMBQ+B £KL¦ £ ©Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.13/37Quadrature Errors3 *PO 32/(where E©R Q+ ¦ £B 7 - "is the quadrature residual.

© £ WWZE]\ £© ¢©\Z]¢7^:W\£,¢ ££ ¦7£ £.Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.14/37The Lorenz System¦¨ZEYXW T S[WEZEZ\ TVUwhere we, as ususal, take, and¦ W_a`^[Y

cebdThe Solution?505040302045403530cements100−10−20252015105−300 10 30 400−20 −10 10 20Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.15/37

;¢7¢¢ ¢) -kkjjiihhlgffcementsA Simple Experiment2020151510105500−5−5−10−10−15−15−2020 25 35 40−2020 25 35 40Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.16/37

;¢7¢¢ ¢) -kkmmiihhlgffcementsA Simple Experiment2020151510105500−5−5−10−10−15−15−2020 25 35 40−2020 25 35 40Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.16/37

;¢7¢¢ ¢) -kknniihhlgffcementsA Simple Experiment2020151510105500−5−5−10−10−15−15−2020 25 35 40−2020 25 35 40Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.16/37

;¢7¢¢ ¢) -kkooiihhlgffcementsA Simple Experiment2020151510105500−5−5−10−10−15−15−2020 25 35 40−2020 25 35 40Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.16/37

;¢7¢¢ ¢) -kkppiihhlgffcementsA Simple Experiment2020151510105500−5−5−10−10−15−15−2020 25 35 40−2020 25 35 40Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.16/37

;¢7¢¢ ¢) -kkqqjjiihhlgffcementsA Simple Experiment2020151510105500−5−5−10−10−15−15−2020 25 35 40−2020 25 35 40Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.16/37

srty wx vH/uThe Residualx 10 −113210lacements−1−2−320 22 24 26 28 32 34 36 38 40[residual with(Pz]Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.17/37

€qj~}|{fode45Trying the same thing with Matlabs ode45, weget the following results:20cements151050−5−10−15−200 10 20 30 40 60 70 80 90 100Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.18/37

€€qj~}|{fode45Trying the same thing with Matlabs ode45, weget the following results:20cements151050−5−10−15−200 10 20 30 40 60 70 80 90 100Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.18/37

‚€qj~}|{fode45Trying the same thing with Matlabs ode45, weget the following results:20cements151050−5−10−15−200 10 20 30 40 60 70 80 90 100Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.18/37

ƒ€qj~}|{fode45Trying the same thing with Matlabs ode45, weget the following results:20cements151050−5−10−15−200 10 20 30 40 60 70 80 90 100Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.18/37

…€qj„m~}|{fode45Trying the same thing with Matlabs ode45, weget the following results:20cements151050−5−10−15−200 10 20 30 40 60 70 80 90 100Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.18/37

7¢& (†-AMGetting FurtherSmall Galerkin errorLarge computational errorWe must thus decrease the computational error,which means decreasing the discrete residual.This in turn means that we have to increase thetimestep!Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.19/37

-7 ¢),kpihgfcementsA Simple Experiment — continuedIncreasing the timestep to we get thefollowing results when increasing the order:20151050−5−10−15−200 10 20 30 40 60 70 80 90 100Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.20/37

-7 ¢),k‡ihgfcementsA Simple Experiment — continuedIncreasing the timestep to we get thefollowing results when increasing the order:20151050−5−10−15−200 10 20 30 40 60 70 80 90 100Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.20/37

-7 ¢),kˆihgfcementsA Simple Experiment — continuedIncreasing the timestep to we get thefollowing results when increasing the order:20151050−5−10−15−200 10 20 30 40 60 70 80 90 100Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.20/37

-7 ¢),k‰ihgfcementsA Simple Experiment — continuedIncreasing the timestep to we get thefollowing results when increasing the order:20151050−5−10−15−200 10 20 30 40 60 70 80 90 100Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.20/37

-7 ¢),kŠihgfcementsA Simple Experiment — continuedIncreasing the timestep to we get thefollowing results when increasing the order:20151050−5−10−15−200 10 20 30 40 60 70 80 90 100Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.20/37

-7 ¢),kqjihgfcementsA Simple Experiment — continuedIncreasing the timestep to we get thefollowing results when increasing the order:20151050−5−10−15−200 10 20 30 40 60 70 80 90 100Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.20/37

-7 ¢),kjjihgfcementsA Simple Experiment — continuedIncreasing the timestep to we get thefollowing results when increasing the order:20151050−5−10−15−200 10 20 30 40 60 70 80 90 100Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.20/37

-7 ¢),kmjihgfcementsA Simple Experiment — continuedIncreasing the timestep to we get thefollowing results when increasing the order:20151050−5−10−15−200 10 20 30 40 60 70 80 90 100Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.20/37

-7 ¢),knjihgfcementsA Simple Experiment — continuedIncreasing the timestep to we get thefollowing results when increasing the order:20151050−5−10−15−200 10 20 30 40 60 70 80 90 100Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.20/37

-7 ¢),kojihgfcementsA Simple Experiment — continuedIncreasing the timestep to we get thefollowing results when increasing the order:20151050−5−10−15−200 10 20 30 40 60 70 80 90 100Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.20/37

-7 ¢),kpjihgfcementsA Simple Experiment — continuedIncreasing the timestep to we get thefollowing results when increasing the order:20151050−5−10−15−200 10 20 30 40 60 70 80 90 100Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.20/37

ŽŒ‹The Stability Factor10 1610 1410 1210 1010 8lacements10 610 410 210 00 5 10 15 20 30 35 40 45 50Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.21/37

k”i“’‘Galerkin Stability Factors10 40 S 210 35S 1S 010 3010 2510 20lacements10 1510 1010 510 00 10 20 30 40 60 70 80 90 100Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.22/37

? •– £¢73¦© M2J•– & †(_- )•– ©¦¢7MAA Simple EstimateA simple estimate for the stability factors isso that7¢& (†-¢7Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.23/37

7—A-7¢¢ ¢),˜``7`-;7 ¢),7`-šHow far we getKeepingwe find the maximum value of :withwithnot much further with larger timestep, sincesoon we will have .Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.24/37

kk””ii€’’‘‘kk””ii€’’‘‘The Approximation10 40 0 20 40 60 80 10010 4010 2010 2010 00 20 40 60 80 10010 010 40 0 20 40 60 80 100lacements10 2010 4010 2010 00 20 40 60 80 10010 0Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.25/37

› œW "/ W"EW››"–¦ W"EW©žThe Solar systemA problem of more practical relevance than theLorenz system:force equals mass times acceleration.Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.26/37

cedŸbThe Earth and the Moon1050−5lacements−100 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5b c60.540−0.510.50−0.5−1−10120−23.2 3.4 3.6 3.8Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.27/37

¢¡¢¡tDistance Earth—Moonx 10 −32.72.652.690 91 92 93 94 95 96 97 98 99 100x 10 −3lacements2.72.652.699 99.1 99.2 99.3 99.4 99.5 99.6 99.7 99.8 99.9 100Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.28/37

£¤¥¤›£›¤›¥ ›¤> £¤_> ¥¤_›£ ›¥¥¤ £¤>¢) MComparable influences Sun/EarthAnders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.29/37

¦fThe Dual10008006004002000−200lacements−400−600−800−10000 1 2 3 4 5 6 7 8 9 10Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.30/37

cbThe Earth, the Moon and the Comet1050−5lacements−10−150 0.2 0.4 0.6 0.8 1 1.2Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.31/37

Live from the set10.50−0.5−11last00.5−1−0.500.5−0.5Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.32/37−1

Live from the set10.50−0.5−110.50−1−0.500.5−0.5Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.32/37−1


































































































Live from the set10.50−0.5−11first00.5−1−0.500.5−0.5Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.32/37−1

§fTimestepsx 10 −310.80.60.4lacements0.200 0.2 0.4 0.6 0.8 1 1.2Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.33/37

¦f¦fThe Dual2 x 10510−1−20 0.2 0.4 0.6 0.8 1 1.26000lacements400020000−2000−4000−60000.91 0.915 0.92 0.925 0.93 0.935 0.94 0.945 0.95 0.955 0.96Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.34/37

Ž Œ © Ž Œ ª Ž ¨Œ‹‹Stability Factors10 510 010 010 −1 10 010 50 0.2 0.4 0.6 0.8 1lacements10 510 010 00 0.2 0.4 0.6 0.8 110 −1 10 010 510 1010 010 00 0.2 0.4 0.6 0.8 110 −1 10 010 10Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.35/37

«««ConclusionsThere are several sources for the total error.Each of these are equally important, in thatfor a certain application any one of these maydominate.From computing the stability factors, we maysee clearly which of the sources is dominantfor a specific problem and tune theimplementation accordingly.Anders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.36/37

«««Chalmers Finite Element CenterVisit our homepage!www.phi.chalmers.sewww.phi.chalmers.se/preprintswww.phi.chalmers.se/presentationswww.phi.chalmers.se/seminarsAnders Logg — Chalmers Finite Element Center — CAM Seminar 2001–04–04 – p.37/37

Multi-adaptive Time-Stepping - FEniCS Project

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