This page demonstrates the temporal convergence rates of advection schemes.
Configurations
A 2D cavity of unit length is used and time step is reduced, as well as the spatial mesh, for testing convergence. The schemes are verified in
- See also
- Implicit schemes comparison and
-
Explicit schemes comparison.
Spatial mesh
The spatial mesh is also refined in order to suppress, as much as possible, the effect of spatial errors. A question far from being trivial, a detailed documentation will be written later on the subject. To sum up, the mesh has to be adapted depending on the spatial order and the temporal order. Also, in order to avoid very large meshes and reduce computational costs, the mesh is adapted for each case.
Spatial schemes
Conjointly with temporal schemes, spatial schemes are also tested in order to test all possible configurations.
Analysis
The precise temporal convergence analysis is somehow complex as spatial errors also occurs and cannot be completely removed. Hence, we have chosen numerical parameters such that the effective schemes' convergence rate appear in the range. Absolute values of error cannot be compared between schemes as the meshes are not the same, for each case.
In order to assess the temporal convergence rate, it is important to understand that the numerical convergence rate - as computed by Notus - is one order lower than the effective temporal convergence rate. In other terms, if the order column shows 1, then the effective temporal order is 2.
- Note
- \(N^{th}\) order WENO schemes should behave like the associated \(N^{th}\) order HOUC schemes in the limit.
Implicit schemes
All schemes exhibit the expected convergence rate except for:
- BDF2+QUICK: this couple fails to convergence at second order and seems to converge at order 1.67.
Explicit schemes
All schemes exhibit the expected convergence rate except:
- Second order RK NSSP3:
- Second order upwind: order 1.5;
- WENO3: recovering order 2 in the limit (above);
- WENO3 FD: barely order 1.7 (absolute error around twice WENO3);
- WENO5: around order 2 in the limit;
- WENO5 FD: around order 2 in the limit (absolute error around thrice WENO5);
- Third order RK NSSP5:
- Second order upwind: order 2;
- WENO3: recovering order 3 in the limit (above);
- WENO3: barely order 2.4 (absolute error around 50 times WENO3 in the limit);
Results
Implicit schemes
First order Euler
o1_upwind
| Time step | Species concentration L1 error 'ink' | order | Species concentration L2 error 'ink' | order | Species concentration Linf error 'ink' | order |
| 0.0004 | 0.002328245593400166 | n/a | 0.0025860193365874044 | n/a | 0.003657162400216185 | n/a |
| 0.0002 | 0.0016279204269153708 | 0.516 | 0.001808189945998648 | 0.516 | 0.002557105398474979 | 0.516 |
| 0.0001 | 0.0011267409074215717 | 0.531 | 0.0012514830298459322 | 0.531 | 0.0017698624634314708 | 0.531 |
| 5e-05 | 0.0008758725783591566 | 0.363 | 0.0009728552547383754 | 0.363 | 0.0013758034247058903 | 0.363 |
| 2.5e-05 | 0.0007503816555253929 | 0.223 | 0.0008334807186122667 | 0.223 | 0.0011786847505178555 | 0.223 |
| 1.25e-05 | 0.0006876336804163223 | 0.126 | 0.000763779283048032 | 0.126 | 0.0010801214649662771 | 0.126 |
| 6.25e-06 | 0.0006562566287869871 | 0.067 | 0.000728925147907609 | 0.067 | 0.0010308350092025353 | 0.067 |
| 3.125e-06 | 0.0006405672006705122 | 0.035 | 0.0007114972415917981 | 0.035 | 0.0010061903640009584 | 0.035 |
| 1.5625e-06 | 0.0006327222641791133 | 0.018 | 0.0007027830806767538 | 0.018 | 0.000993867692014927 | 0.018 |
| 7.8125e-07 | 0.000628799740675303 | 0.009 | 0.0006984259484739052 | 0.009 | 0.000987706269241495 | 0.009 |
BDF 2
o1_upwind
| Time step | Species concentration L1 error 'ink' | order | Species concentration L2 error 'ink' | order | Species concentration Linf error 'ink' | order |
| 0.0004 | 0.016897161553552656 | n/a | 0.018765067486536512 | n/a | 0.02653064284104778 | n/a |
| 0.0002 | 0.008331502962727642 | 1.020 | 0.009260199166304817 | 1.019 | 0.013081494707425478 | 1.020 |
| 0.0001 | 0.004081846450255854 | 1.029 | 0.004533683182238322 | 1.030 | 0.006411580661454197 | 1.029 |
| 5e-05 | 0.002026058564239701 | 1.011 | 0.002250353608635179 | 1.011 | 0.0031824415905300496 | 1.011 |
| 2.5e-05 | 0.001009184297075404 | 1.005 | 0.0011209348393920165 | 1.005 | 0.0015851812658380648 | 1.005 |
| 1.25e-05 | 0.000503629058660489 | 1.003 | 0.0005593928129382993 | 1.003 | 0.0007910994011496308 | 1.003 |
| 6.25e-06 | 0.00025157165337527 | 1.001 | 0.0002794260460263367 | 1.001 | 0.00039516791500537884 | 1.001 |
| 3.125e-06 | 0.0001257248508933426 | 1.001 | 0.00013964522287961657 | 1.001 | 0.00019748814443032447 | 1.001 |
| 1.5625e-06 | 6.284714983895403e-05 | 1.000 | 6.980563544144747e-05 | 1.000 | 9.872007342059952e-05 | 1.000 |
| 7.8125e-07 | 3.1419752182194546e-05 | 1.000 | 3.4898570599427174e-05 | 1.000 | 4.935403152894757e-05 | 1.000 |
o2_upwind
| Time step | Species concentration L1 error 'ink' | order | Species concentration L2 error 'ink' | order | Species concentration Linf error 'ink' | order |
| 0.0004 | 0.0011202383815321275 | n/a | 0.0012441448822416454 | n/a | 0.0017588435739220865 | n/a |
| 0.0002 | 0.00034851284947976934 | 1.685 | 0.0003870750270259119 | 1.684 | 0.0005474066322880278 | 1.684 |
| 0.0001 | 0.00011267666878853863 | 1.629 | 0.0001251384698988966 | 1.629 | 0.00017693107501426386 | 1.629 |
| 5e-05 | 4.55854484824282e-05 | 1.306 | 5.0632590826027994e-05 | 1.305 | 7.160516199883338e-05 | 1.305 |
| 2.5e-05 | 2.135430672298822e-05 | 1.094 | 2.3718181896925282e-05 | 1.094 | 3.354118718451593e-05 | 1.094 |
| 1.25e-05 | 1.0495387781963584e-05 | 1.025 | 1.165751679939978e-05 | 1.025 | 1.648604911033691e-05 | 1.025 |
| 6.25e-06 | 5.240351593556918e-06 | 1.002 | 5.820563536788645e-06 | 1.002 | 8.231479801085184e-06 | 1.002 |
| 3.125e-06 | 2.616564407561554e-06 | 1.002 | 2.906267906062392e-06 | 1.002 | 4.110072941432286e-06 | 1.002 |
| 1.5625e-06 | 1.30906520342613e-06 | 0.999 | 1.454005819193909e-06 | 0.999 | 2.0562745502682167e-06 | 0.999 |
| 7.8125e-07 | 6.545461367216091e-07 | 1.000 | 7.270191991877827e-07 | 1.000 | 1.0281604013555323e-06 | 1.000 |
o2_centered
| Time step | Species concentration L1 error 'ink' | order | Species concentration L2 error 'ink' | order | Species concentration Linf error 'ink' | order |
| 0.0004 | 0.0010595163689997422 | n/a | 0.0011763771582278082 | n/a | 0.0016623690348018805 | n/a |
| 0.0002 | 0.0003146820766263017 | 1.751 | 0.00034956338959988447 | 1.751 | 0.0004942688800808615 | 1.750 |
| 0.0001 | 8.803022875185452e-05 | 1.838 | 9.776494276630402e-05 | 1.838 | 0.00013822036757527023 | 1.838 |
| 5e-05 | 2.890871064975307e-05 | 1.606 | 3.210937527997729e-05 | 1.606 | 4.540951088116696e-05 | 1.606 |
| 2.5e-05 | 1.173127395565095e-05 | 1.301 | 1.303007493725124e-05 | 1.301 | 1.8426873025834123e-05 | 1.301 |
| 1.25e-05 | 5.431877479894099e-06 | 1.111 | 6.033316763102866e-06 | 1.111 | 8.532338296979303e-06 | 1.111 |
| 6.25e-06 | 2.655334865221735e-06 | 1.033 | 2.9493116775374926e-06 | 1.033 | 4.170867940411771e-06 | 1.033 |
| 3.125e-06 | 1.315682984224862e-06 | 1.013 | 1.461366243316009e-06 | 1.013 | 2.0666615785439046e-06 | 1.013 |
| 1.5625e-06 | 6.562167892872227e-07 | 1.004 | 7.288735963697946e-07 | 1.004 | 1.0307828779317074e-06 | 1.004 |
| 7.8125e-07 | 3.2767395685411946e-07 | 1.002 | 3.6395489492522313e-07 | 1.002 | 5.147100072076111e-07 | 1.002 |
QUICK
| Time step | Species concentration L1 error 'ink' | order | Species concentration L2 error 'ink' | order | Species concentration Linf error 'ink' | order |
| 0.0004 | 0.0010805751212866047 | n/a | 0.0012026542377071832 | n/a | 0.0016949491932104488 | n/a |
| 0.0002 | 0.00032726198986172766 | 1.723 | 0.00036347745444074156 | 1.726 | 0.0005140326726411626 | 1.721 |
| 0.0001 | 9.65432903109697e-05 | 1.761 | 0.0001072567645940209 | 1.761 | 0.00015156431363572143 | 1.762 |
| 5e-05 | 3.732600805884568e-05 | 1.371 | 4.1455581627366606e-05 | 1.371 | 5.8612092800891524e-05 | 1.371 |
| 2.5e-05 | 1.9493339510505658e-05 | 0.937 | 2.1651559939334553e-05 | 0.937 | 3.0619635505102494e-05 | 0.937 |
| 1.25e-05 | 1.1572935638302587e-05 | 0.752 | 1.2852594708030956e-05 | 0.752 | 1.8170637561792935e-05 | 0.753 |
| 6.25e-06 | 7.182431772304141e-06 | 0.688 | 7.977964566094953e-06 | 0.688 | 1.1282246608651514e-05 | 0.688 |
| 3.125e-06 | 4.492926841462474e-06 | 0.677 | 4.990370073986383e-06 | 0.677 | 7.057448311295622e-06 | 0.677 |
| 1.5625e-06 | 2.826318579559108e-06 | 0.669 | 3.13935396432197e-06 | 0.669 | 4.439700273646541e-06 | 0.669 |
| 7.8125e-07 | 1.7807280584853974e-06 | 0.666 | 1.9778629491277024e-06 | 0.667 | 2.7970398907978256e-06 | 0.667 |
HOUC3
| Time step | Species concentration L1 error 'ink' | order | Species concentration L2 error 'ink' | order | Species concentration Linf error 'ink' | order |
| 0.0004 | 0.001087273094510052 | n/a | 0.0012062103604954593 | n/a | 0.0017054553802297034 | n/a |
| 0.0002 | 0.00032199681928483833 | 1.756 | 0.0003576284027758857 | 1.754 | 0.0005057626328051068 | 1.754 |
| 0.0001 | 8.589111086407597e-05 | 1.906 | 9.550151896826426e-05 | 1.905 | 0.00013484134655095126 | 1.907 |
| 5e-05 | 2.4139749605240727e-05 | 1.831 | 2.6804494432507563e-05 | 1.833 | 3.788328713949518e-05 | 1.832 |
| 2.5e-05 | 7.363729523684143e-06 | 1.713 | 8.178992849127228e-06 | 1.712 | 1.1566756568326397e-05 | 1.712 |
| 1.25e-05 | 2.492181110834756e-06 | 1.563 | 2.7678847885368665e-06 | 1.563 | 3.913717120274107e-06 | 1.563 |
| 6.25e-06 | 9.5207456266122e-07 | 1.388 | 1.0575467253193524e-06 | 1.388 | 1.4955685992967815e-06 | 1.388 |
| 3.125e-06 | 4.0149209368500306e-07 | 1.246 | 4.459464631391257e-07 | 1.246 | 6.30660094258495e-07 | 1.246 |
| 1.5625e-06 | 1.8250507353750263e-07 | 1.137 | 2.0270698937574636e-07 | 1.137 | 2.8665394247173737e-07 | 1.138 |
| 7.8125e-07 | 8.677789946586157e-08 | 1.073 | 9.63840344281785e-08 | 1.073 | 1.3630032202893716e-07 | 1.073 |
Explicit schemes
First order Euler
o1_upwind
| Time step | Species concentration L1 error 'ink' | order | Species concentration L2 error 'ink' | order | Species concentration Linf error 'ink' | order |
| 0.0004 | 0.03628566220965192 | n/a | 0.042508904749132166 | n/a | 0.054428493314477966 | n/a |
| 0.0002 | 0.03657152982385984 | -0.011 | 0.043047227855582536 | -0.018 | 0.05485729473578993 | -0.011 |
| 0.0001 | 0.03676636621494176 | -0.008 | 0.043425004436982766 | -0.013 | 0.05514954932241278 | -0.008 |
| 5e-05 | 0.0368611062377317 | -0.004 | 0.04361199728635508 | -0.006 | 0.055291659356597855 | -0.004 |
| 2.5e-05 | 0.036907813186603286 | -0.002 | 0.043705022023143865 | -0.003 | 0.055361719779904894 | -0.002 |
| 1.25e-05 | 0.036931001713603404 | -0.001 | 0.043751416748765246 | -0.002 | 0.05539650257040474 | -0.001 |
| 6.25e-06 | 0.03694255484299902 | -0.000 | 0.04377458473624694 | -0.001 | 0.05541383226449914 | -0.000 |
| 3.125e-06 | 0.03694832113704307 | -0.000 | 0.043786161390603526 | -0.000 | 0.05542248170556485 | -0.000 |
| 1.5625e-06 | 0.03695120171801226 | -0.000 | 0.04379194788349368 | -0.000 | 0.05542680257701749 | -0.000 |
| 7.8125e-07 | 0.03695264136718522 | -0.000 | 0.04379484067143757 | -0.000 | 0.05542896205078052 | -0.000 |
Second order RK NSSP3
First order upwind
| Time step | Species concentration L1 error 'ink' | order | Species concentration L2 error 'ink' | order | Species concentration Linf error 'ink' | order |
| 0.0004 | 0.03200784187800361 | n/a | 0.03560940271321561 | n/a | 0.05018595786032576 | n/a |
| 0.0002 | 0.01594632951284059 | 1.005 | 0.017704623231781135 | 1.008 | 0.02503771841146185 | 1.003 |
| 0.0001 | 0.00804367654541616 | 0.987 | 0.008937884255019525 | 0.986 | 0.01262957147441568 | 0.987 |
| 5e-05 | 0.004007519849337339 | 1.005 | 0.004451159308135382 | 1.006 | 0.0062948317825128886 | 1.005 |
| 2.5e-05 | 0.002007342922027756 | 0.997 | 0.002229558036905677 | 0.997 | 0.00315304390222082 | 0.997 |
| 1.25e-05 | 0.0010044899787567546 | 0.999 | 0.0011157175735087828 | 0.999 | 0.0015778076419354914 | 0.999 |
Second order upwind
| Time step | Species concentration L1 error 'ink' | order | Species concentration L2 error 'ink' | order | Species concentration Linf error 'ink' | order |
| 0.0004 | 0.06339248056391039 | n/a | 0.06982990065196364 | n/a | 0.09874307786622227 | n/a |
| 0.0002 | 0.04865423508575584 | 0.382 | 0.054006737230684866 | 0.371 | 0.07633886286893521 | 0.371 |
| 0.0001 | 0.03446801350660621 | 0.497 | 0.03798436080314233 | 0.508 | 0.05368898189976157 | 0.508 |
| 5e-05 | 0.024987196753024014 | 0.464 | 0.027756099826995756 | 0.453 | 0.039239185750417405 | 0.452 |
| 2.5e-05 | 0.017622596833637115 | 0.504 | 0.019555901693326657 | 0.505 | 0.027600435739204454 | 0.508 |
| 1.25e-05 | 0.012564703675519307 | 0.488 | 0.013955945421285717 | 0.487 | 0.019735289918181187 | 0.484 |
HOUC3
| Time step | Species concentration L1 error 'ink' | order | Species concentration L2 error 'ink' | order | Species concentration Linf error 'ink' | order |
| 0.0004 | 4.227884023518123e-05 | n/a | 4.689782689232608e-05 | n/a | 6.631698688486232e-05 | n/a |
| 0.0002 | 2.1551024298687107e-05 | 0.972 | 2.3939806784324317e-05 | 0.970 | 3.385034303160506e-05 | 0.970 |
| 0.0001 | 1.0485653833166386e-05 | 1.039 | 1.1658216642810617e-05 | 1.038 | 1.646153680068263e-05 | 1.040 |
| 5e-05 | 5.25671403771473e-06 | 0.996 | 5.838100988023235e-06 | 0.998 | 8.253940833813367e-06 | 0.996 |
| 2.5e-05 | 2.6446526492899753e-06 | 0.991 | 2.937431013786449e-06 | 0.991 | 4.1541522440846634e-06 | 0.991 |
| 1.25e-05 | 1.312116653023084e-06 | 1.011 | 1.4576672303172206e-06 | 1.011 | 2.0614402143070976e-06 | 1.011 |
WENO3
| Time step | Species concentration L1 error 'ink' | order | Species concentration L2 error 'ink' | order | Species concentration Linf error 'ink' | order |
| 0.0004 | 0.0005077723604586756 | n/a | 0.0010509481406723133 | n/a | 0.003875114673755675 | n/a |
| 0.0002 | 0.0002969827836759603 | 0.774 | 0.0006142440086219349 | 0.775 | 0.002369848877864844 | 0.709 |
| 0.0001 | 0.00015093800892046358 | 0.976 | 0.00032840516181410973 | 0.903 | 0.001144617720244434 | 1.050 |
| 5e-05 | 7.092622018382947e-05 | 1.090 | 0.00016944121120477184 | 0.955 | 0.0007274733110088638 | 0.654 |
| 2.5e-05 | 3.495414662593043e-05 | 1.021 | 8.524727994456081e-05 | 0.991 | 0.0004055324658831516 | 0.843 |
| 1.25e-05 | 1.5755215153563687e-05 | 1.150 | 4.046984492196087e-05 | 1.075 | 0.0002012783687632691 | 1.011 |
| 6.25e-06 | 7.189686517580275e-06 | 1.132 | 1.876531149843038e-05 | 1.109 | 9.668794357076127e-05 | 1.058 |
| 3.125e-06 | 3.1284453997811e-06 | 1.200 | 8.385504578424358e-06 | 1.162 | 4.4594436907963164e-05 | 1.116 |
| 1.5625e-06 | 1.3574172294807501e-06 | 1.205 | 3.6719235014543323e-06 | 1.191 | 1.953822616351797e-05 | 1.191 |
| 7.8125e-07 | 5.780457897720856e-07 | 1.232 | 1.5789342671414451e-06 | 1.218 | 8.322534108207158e-06 | 1.231 |
WENO3 FD
| Time step | Species concentration L1 error 'ink' | order | Species concentration L2 error 'ink' | order | Species concentration Linf error 'ink' | order |
| 0.0004 | 0.00040049062238167233 | n/a | 0.000520659552317158 | n/a | 0.001214597777762172 | n/a |
| 0.0002 | 0.00022762005922853223 | 0.815 | 0.0003163795137128914 | 0.719 | 0.0008077820129190338 | 0.588 |
| 0.0001 | 0.00012178053456672036 | 0.902 | 0.00017718862284952597 | 0.836 | 0.0004613868316233338 | 0.808 |
| 5e-05 | 6.69056500783368e-05 | 0.864 | 0.00010206537886667384 | 0.796 | 0.0003067393156366005 | 0.589 |
| 2.5e-05 | 3.751256442886491e-05 | 0.835 | 5.9564488202355774e-05 | 0.777 | 0.00019867360973216397 | 0.627 |
| 1.25e-05 | 2.0028876137353543e-05 | 0.905 | 3.358837856739702e-05 | 0.826 | 0.00012071968030930025 | 0.719 |
| 6.25e-06 | 1.0980472608241841e-05 | 0.867 | 1.9188686041439257e-05 | 0.808 | 7.232448538053993e-05 | 0.739 |
| 3.125e-06 | 5.8793793024356024e-06 | 0.901 | 1.0969331761365826e-05 | 0.807 | 4.810423514482742e-05 | 0.588 |
| 1.5625e-06 | 3.120014544716256e-06 | 0.914 | 6.212125154109537e-06 | 0.820 | 2.9414183152096096e-05 | 0.710 |
| 7.8125e-07 | 1.6716507326991618e-06 | 0.900 | 3.5317790307265956e-06 | 0.815 | 1.7868748770147325e-05 | 0.719 |
HOUC5
| Time step | Species concentration L1 error 'ink' | order | Species concentration L2 error 'ink' | order | Species concentration Linf error 'ink' | order |
| 0.0004 | 7.738037199341271e-05 | n/a | 8.580743691754899e-05 | n/a | 0.00012113612640540783 | n/a |
| 0.0002 | 3.400749984654706e-05 | 1.186 | 3.7698314765438504e-05 | 1.187 | 5.328896477696521e-05 | 1.185 |
| 0.0001 | 1.6759613776811768e-05 | 1.021 | 1.858203785763666e-05 | 1.021 | 2.627785010134165e-05 | 1.020 |
| 5e-05 | 9.012647851939612e-06 | 0.895 | 9.996522432994767e-06 | 0.894 | 1.4136897939831528e-05 | 0.894 |
| 2.5e-05 | 4.556016068931376e-06 | 0.984 | 5.059937638562713e-06 | 0.982 | 7.154638529516255e-06 | 0.983 |
| 1.25e-05 | 2.236958148384226e-06 | 1.026 | 2.484132357730715e-06 | 1.026 | 3.513092121654182e-06 | 1.026 |
WENO5
| Time step | Species concentration L1 error 'ink' | order | Species concentration L2 error 'ink' | order | Species concentration Linf error 'ink' | order |
| 0.0004 | 0.00018733568879110047 | n/a | 0.00023369636134966588 | n/a | 0.0005045355271577345 | n/a |
| 0.0002 | 8.75588906177434e-05 | 1.097 | 0.00010572440296991992 | 1.144 | 0.0002205304137254116 | 1.194 |
| 0.0001 | 4.52065698916912e-05 | 0.954 | 5.267947600434535e-05 | 1.005 | 0.00010431734797722303 | 1.080 |
| 5e-05 | 2.503588850866947e-05 | 0.853 | 2.825529398488545e-05 | 0.899 | 5.220697020380527e-05 | 0.999 |
| 2.5e-05 | 1.2750261514794627e-05 | 0.973 | 1.4167033288853595e-05 | 0.996 | 2.3569827776293018e-05 | 1.147 |
| 1.25e-05 | 6.296438026624428e-06 | 1.018 | 6.800107650380298e-06 | 1.059 | 9.918241160644659e-06 | 1.249 |
| 6.25e-06 | 2.9288485126040772e-06 | 1.104 | 3.228791653417023e-06 | 1.075 | 4.739865897995799e-06 | 1.065 |
| 3.125e-06 | 1.4354648753047222e-06 | 1.029 | 1.554140787872663e-06 | 1.055 | 2.509562266994614e-06 | 0.917 |
| 1.5625e-06 | 7.145445488674652e-07 | 1.006 | 7.571938034366878e-07 | 1.037 | 1.2976611679649253e-06 | 0.952 |
| 7.8125e-07 | 3.5720365592803817e-07 | 1.000 | 3.7970986067571066e-07 | 0.996 | 6.845183795345378e-07 | 0.923 |
WENO5 FD
| Time step | Species concentration L1 error 'ink' | order | Species concentration L2 error 'ink' | order | Species concentration Linf error 'ink' | order |
| 0.0004 | 0.000752429601203556 | n/a | 0.0008368624613106343 | n/a | 0.0012944969982995325 | n/a |
| 0.0002 | 0.00032791040341823436 | 1.198 | 0.00037125580880534987 | 1.173 | 0.000593865848277142 | 1.124 |
| 0.0001 | 0.0001586084240616107 | 1.048 | 0.00018077485227087426 | 1.038 | 0.00029215163073026407 | 1.023 |
| 5e-05 | 8.416227435652338e-05 | 0.914 | 9.540511563449772e-05 | 0.922 | 0.0001551714582858077 | 0.913 |
| 2.5e-05 | 4.137669868244117e-05 | 1.024 | 4.7102770685530175e-05 | 1.018 | 7.649823453903526e-05 | 1.020 |
| 1.25e-05 | 1.9793547395880714e-05 | 1.064 | 2.2504161619050774e-05 | 1.066 | 3.7022084838245206e-05 | 1.047 |
| 6.25e-06 | 9.576391473103086e-06 | 1.047 | 1.0681642448086663e-05 | 1.075 | 1.731113689651398e-05 | 1.097 |
| 3.125e-06 | 4.566733926835975e-06 | 1.068 | 5.120095943506529e-06 | 1.061 | 8.357577225837076e-06 | 1.051 |
| 1.5625e-06 | 2.1619200581270284e-06 | 1.079 | 2.48231167268698e-06 | 1.044 | 4.381422842314464e-06 | 0.932 |
| 7.8125e-07 | 1.0781482161552388e-06 | 1.004 | 1.2382779794274354e-06 | 1.003 | 2.2304704143660814e-06 | 0.974 |
Third order RK NSSP5
First order upwind
| Time step | Species concentration L1 error 'ink' | order | Species concentration L2 error 'ink' | order | Species concentration Linf error 'ink' | order |
| 0.0004 | 0.013066568814402966 | n/a | 0.014498064482381872 | n/a | 0.020501545304989377 | n/a |
| 0.0002 | 0.0032341704056976566 | 2.014 | 0.0035921993016912912 | 2.013 | 0.005080136036277327 | 2.013 |
| 0.0001 | 0.000808905924535171 | 1.999 | 0.0008984688577613584 | 1.999 | 0.0012706251000331736 | 1.999 |
| 5e-05 | 0.00020216222052607733 | 2.000 | 0.00022454571954212322 | 2.000 | 0.0003175555888978998 | 2.000 |
| 2.5e-05 | 5.053130526714601e-05 | 2.000 | 5.612616850042102e-05 | 2.000 | 7.937438837168909e-05 | 2.000 |
| 1.25e-05 | 1.2633201968572557e-05 | 2.000 | 1.4031959343652468e-05 | 2.000 | 1.9844187165718452e-05 | 2.000 |
Second order upwind
| Time step | Species concentration L1 error 'ink' | order | Species concentration L2 error 'ink' | order | Species concentration Linf error 'ink' | order |
| 0.0004 | 0.059370051075927036 | n/a | 0.06603460805806534 | n/a | 0.09308790304633119 | n/a |
| 0.0002 | 0.031081370085307446 | 0.934 | 0.034527839004295235 | 0.935 | 0.048801612396866534 | 0.932 |
| 0.0001 | 0.01594629090446416 | 0.963 | 0.01770458017982573 | 0.964 | 0.025037657791514678 | 0.963 |
| 5e-05 | 0.007983242368609892 | 0.998 | 0.008866982039647756 | 0.998 | 0.012539717750302826 | 0.998 |
| 2.5e-05 | 0.004007519064400671 | 0.994 | 0.004451158436917768 | 0.994 | 0.006294830549568697 | 0.994 |
| 1.25e-05 | 0.0020072893204612008 | 0.997 | 0.002229557926450552 | 0.997 | 0.0031529597072927906 | 0.997 |
HOUC3
| Time step | Species concentration L1 error 'ink' | order | Species concentration L2 error 'ink' | order | Species concentration Linf error 'ink' | order |
| 0.0004 | 0.00013513661460938783 | n/a | 0.0001500495632544096 | n/a | 0.0002121071204438385 | n/a |
| 0.0002 | 3.3934769639927016e-05 | 1.994 | 3.768915618521876e-05 | 1.993 | 5.330049813367932e-05 | 1.993 |
| 0.0001 | 8.462974326674867e-06 | 2.004 | 9.406094503637791e-06 | 2.002 | 1.3287109610238268e-05 | 2.004 |
| 5e-05 | 2.123131719378828e-06 | 1.995 | 2.358064751736055e-06 | 1.996 | 3.3342812032488567e-06 | 1.995 |
| 2.5e-05 | 5.312920817215231e-07 | 1.999 | 5.901275425872623e-07 | 1.999 | 8.344872510779311e-07 | 1.998 |
| 1.25e-05 | 1.3242816119582536e-07 | 2.004 | 1.4708629550718457e-07 | 2.004 | 2.0799529121262594e-07 | 2.004 |
WENO3
| Time step | Species concentration L1 error 'ink' | order | Species concentration L2 error 'ink' | order | Species concentration Linf error 'ink' | order |
| 0.0004 | 0.0014191407269093998 | n/a | 0.002499901602068865 | n/a | 0.008726577595501772 | n/a |
| 0.0002 | 0.00044934796776327404 | 1.659 | 0.0008981848059147519 | 1.477 | 0.003301812148932437 | 1.402 |
| 0.0001 | 0.00011947116617365731 | 1.911 | 0.000270058592830414 | 1.734 | 0.0009427282180234364 | 1.808 |
| 5e-05 | 2.7633859335091778e-05 | 2.112 | 6.791117816977718e-05 | 1.992 | 0.00032028878398360217 | 1.557 |
| 2.5e-05 | 5.567083263663519e-06 | 2.311 | 1.470621524715361e-05 | 2.207 | 7.219166227900864e-05 | 2.149 |
| 1.25e-05 | 1.0400312401475752e-06 | 2.420 | 2.8314251574720614e-06 | 2.377 | 1.5010370389423322e-05 | 2.266 |
| 6.25e-06 | 1.836787794454237e-07 | 2.501 | 5.069372325819415e-07 | 2.482 | 2.750069762025298e-06 | 2.448 |
| 3.125e-06 | 3.1642367103012336e-08 | 2.537 | 8.830239291986768e-08 | 2.521 | 5.236694836652234e-07 | 2.393 |
| 1.5625e-06 | 5.453365775462358e-09 | 2.537 | 1.4946079180530712e-08 | 2.563 | 9.46461117257158e-08 | 2.468 |
| 7.8125e-07 | 9.670312202050825e-10 | 2.496 | 2.4864507271359996e-09 | 2.588 | 1.58081318035741e-08 | 2.582 |
WENO3 FD
| Time step | Species concentration L1 error 'ink' | order | Species concentration L2 error 'ink' | order | Species concentration Linf error 'ink' | order |
| 0.0004 | 0.0010832373838219 | n/a | 0.0013322414195814484 | n/a | 0.0029224716088314118 | n/a |
| 0.0002 | 0.0003358461837305344 | 1.689 | 0.00045325724054943176 | 1.555 | 0.0010918794955941946 | 1.420 |
| 0.0001 | 0.00010055380263571706 | 1.740 | 0.0001494430887591597 | 1.601 | 0.0004082780186191459 | 1.419 |
| 5e-05 | 3.096918093225543e-05 | 1.699 | 4.942586181270677e-05 | 1.596 | 0.0001581259937536439 | 1.368 |
| 2.5e-05 | 9.080024585910467e-06 | 1.770 | 1.6152123370999865e-05 | 1.614 | 6.490847612186657e-05 | 1.285 |
| 1.25e-05 | 2.5977170755993016e-06 | 1.805 | 5.214159682437044e-06 | 1.631 | 2.5119632127409974e-05 | 1.370 |
| 6.25e-06 | 7.307804024311935e-07 | 1.830 | 1.67431810927853e-06 | 1.639 | 1.0067101073499285e-05 | 1.319 |
| 3.125e-06 | 2.0505366280049272e-07 | 1.833 | 5.369721516231587e-07 | 1.641 | 3.864603407517653e-06 | 1.381 |
| 1.5625e-06 | 5.684803795255091e-08 | 1.851 | 1.7145624059747022e-07 | 1.647 | 1.4847569580790643e-06 | 1.380 |
| 7.8125e-07 | 1.554702913680838e-08 | 1.870 | 5.448317769606176e-08 | 1.654 | 5.602082553853027e-07 | 1.406 |
HOUC5
| Time step | Species concentration L1 error 'ink' | order | Species concentration L2 error 'ink' | order | Species concentration Linf error 'ink' | order |
| 0.0004 | 3.6219177661490463e-06 | n/a | 4.022364584592168e-06 | n/a | 5.687903409934414e-06 | n/a |
| 0.0002 | 9.147431597904009e-07 | 1.985 | 1.0166798194897507e-06 | 1.984 | 1.4360647881961341e-06 | 1.986 |
| 0.0001 | 2.1675211750372185e-07 | 2.077 | 2.4052251325048686e-07 | 2.080 | 3.394971044290429e-07 | 2.081 |
| 5e-05 | 5.4056606437080435e-08 | 2.004 | 6.00390596503841e-08 | 2.002 | 8.490796854498939e-08 | 1.999 |
| 2.5e-05 | 1.3695200979569366e-08 | 1.981 | 1.5211357109180007e-08 | 1.981 | 2.1511800563089878e-08 | 1.981 |
| 1.25e-05 | 3.3148997146007574e-09 | 2.047 | 3.6819847615360143e-09 | 2.047 | 5.2067007372258445e-09 | 2.047 |
WENO5
| Time step | Species concentration L1 error 'ink' | order | Species concentration L2 error 'ink' | order | Species concentration Linf error 'ink' | order |
| 0.0004 | 9.957906084571335e-06 | n/a | 1.0945808828931157e-05 | n/a | 1.7477298630053095e-05 | n/a |
| 0.0002 | 2.5097171765542996e-06 | 1.988 | 2.6535905406489516e-06 | 2.044 | 4.1628054183728835e-06 | 2.070 |
| 0.0001 | 5.865324006013721e-07 | 2.097 | 6.053624389864841e-07 | 2.132 | 7.379391734341212e-07 | 2.496 |
| 5e-05 | 1.3745509141058026e-07 | 2.093 | 1.4848153251001785e-07 | 2.028 | 2.7637423871684774e-07 | 1.417 |
| 2.5e-05 | 3.334523148722599e-08 | 2.043 | 3.68465641071611e-08 | 2.011 | 7.381908206927079e-08 | 1.905 |
| 1.25e-05 | 7.775902166264076e-09 | 2.100 | 8.707590680264218e-09 | 2.081 | 1.720544273720037e-08 | 2.101 |
| 6.25e-06 | 1.8734249361589657e-09 | 2.053 | 2.123684677910971e-09 | 2.036 | 4.424483224241271e-09 | 1.959 |
| 3.125e-06 | 4.5467727862844294e-10 | 2.043 | 5.171913910096274e-10 | 2.038 | 1.0981930920195282e-09 | 2.010 |
| 1.5625e-06 | 1.1152128716295049e-10 | 2.028 | 1.2681418578411912e-10 | 2.028 | 2.84848145071237e-10 | 1.947 |
| 7.8125e-07 | 2.7305470594922652e-11 | 2.030 | 3.096564865462696e-11 | 2.034 | 7.200573470811378e-11 | 1.984 |
WENO5 FD
| Time step | Species concentration L1 error 'ink' | order | Species concentration L2 error 'ink' | order | Species concentration Linf error 'ink' | order |
| 0.0004 | 3.204127592714386e-05 | n/a | 3.644099789849359e-05 | n/a | 5.946171816917367e-05 | n/a |
| 0.0002 | 7.623066506470603e-06 | 2.071 | 8.78862736289884e-06 | 2.052 | 1.4930115052713688e-05 | 1.994 |
| 0.0001 | 1.7207666742472376e-06 | 2.147 | 1.985296741016197e-06 | 2.146 | 3.514545207750075e-06 | 2.087 |
| 5e-05 | 4.1902548096279353e-07 | 2.038 | 4.81329915922731e-07 | 2.044 | 8.888302570619899e-07 | 1.983 |
| 2.5e-05 | 1.0314497509233252e-07 | 2.022 | 1.1846868602467867e-07 | 2.023 | 2.2751397876774604e-07 | 1.966 |
| 1.25e-05 | 2.4353557046269114e-08 | 2.082 | 2.7975332131200775e-08 | 2.082 | 5.355792154215777e-08 | 2.087 |
| 6.25e-06 | 6.0238666014613555e-09 | 2.015 | 6.864776537770281e-09 | 2.027 | 1.3729924863703502e-08 | 1.964 |
| 3.125e-06 | 1.487728909077764e-09 | 2.018 | 1.689159085577286e-09 | 2.023 | 3.4150836603075163e-09 | 2.007 |
| 1.5625e-06 | 3.7036942747766573e-10 | 2.006 | 4.191365203668635e-10 | 2.011 | 8.570840703825411e-10 | 1.994 |
| 7.8125e-07 | 9.182910750791698e-11 | 2.012 | 1.035507088506657e-10 | 2.017 | 2.1189316967706873e-10 | 2.016 |
References
1.11.0