This page demonstrates the spatial convergence rates of advection schemes.
Configurations
A 2D cavity of unit length is used and the spatial mesh is reduced, at constant time step, for testing convergence. The schemes are verified in
- See also
- Implicit schemes comparison and
-
Explicit schemes comparison.
Temporal schemes
In order to suppress most temporal errors, the most precise temporal scheme has to be used.
- All explicit schemes use the third order NSSP5_O3 scheme;
- All implicit schemes use the second order BDF scheme.
Analysis
The precise spatial convergence analysis is somehow complex as temporal 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 can be compared between schemes.
- 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. For the third order schemes (HOUC3 and QUICK), the late convergence rates are decreasing because of the relatively increasing temporal error (second order BDF).
Explicit schemes
All schemes exhibit the expected convergence rate except:
- WENO3: starts with a higher error than HOUC3 and WENO3_FD but ends up recovering its rate (showing super convergence up to the finest mesh) towards the optimal HOUC3;
- WENO3_FD: starts with a smaller error than WENO3 but ends up showing second order convergence, as expected by its FD definition.
Results
Implicit schemes
o1_upwind
| Mesh | Species concentration L1 error 'ink' | order | Species concentration L2 error 'ink' | order | Species concentration Linf error 'ink' | order |
| 32×4 | 0.00015603014623135833 | n/a | 0.00017370722050959214 | n/a | 0.0002451517649921797 | n/a |
| 64×8 | 7.840504957651708e-05 | 0.993 | 8.714004075232867e-05 | 0.995 | 0.00012317358139601886 | 0.993 |
| 128×16 | 3.925398486434204e-05 | 0.998 | 4.3607886495033425e-05 | 0.999 | 6.166383064187819e-05 | 0.998 |
| 256×32 | 1.9634971429621702e-05 | 0.999 | 2.1810167177667102e-05 | 1.000 | 3.084349562754163e-05 | 0.999 |
| 512×64 | 9.81975953045699e-06 | 1.000 | 1.0907222997222287e-05 | 1.000 | 1.5425080947095715e-05 | 1.000 |
| 1024×128 | 4.911373709373294e-06 | 1.000 | 5.4552085133012335e-06 | 1.000 | 7.714827494842424e-06 | 1.000 |
o2_upwind
| Mesh | Species concentration L1 error 'ink' | order | Species concentration L2 error 'ink' | order | Species concentration Linf error 'ink' | order |
| 32×4 | 4.075985369626778e-05 | n/a | 4.519495511872003e-05 | n/a | 6.36001868366265e-05 | n/a |
| 64×8 | 1.0258872635520019e-05 | 1.990 | 1.1389335337312484e-05 | 1.988 | 1.6086339539186922e-05 | 1.983 |
| 128×16 | 2.569103334583954e-06 | 1.998 | 2.8531152744778303e-06 | 1.997 | 4.033458884979135e-06 | 1.996 |
| 256×32 | 6.425999368368476e-07 | 1.999 | 7.136938527179463e-07 | 1.999 | 1.0091564073208548e-06 | 1.999 |
| 512×64 | 1.6069542054095157e-07 | 2.000 | 1.7849614157795538e-07 | 1.999 | 2.524316560226225e-07 | 1.999 |
| 1024×128 | 4.02742087010484e-08 | 1.996 | 4.473352789185852e-08 | 1.996 | 6.326265429568156e-08 | 1.996 |
o2_centered
| Mesh | Species concentration L1 error 'ink' | order | Species concentration L2 error 'ink' | order | Species concentration Linf error 'ink' | order |
| 32×4 | 2.0535398546135675e-05 | n/a | 2.266282891681495e-05 | n/a | 3.144119449810523e-05 | n/a |
| 64×8 | 5.138770573071948e-06 | 1.999 | 5.698576159744286e-06 | 1.992 | 8.020772138253585e-06 | 1.971 |
| 128×16 | 1.2850052843141227e-06 | 2.000 | 1.4267094090068736e-06 | 1.998 | 2.0153249198975876e-06 | 1.993 |
| 256×32 | 3.2127423410391667e-07 | 2.000 | 3.5681934082695544e-07 | 1.999 | 5.045427371383471e-07 | 1.998 |
| 512×64 | 8.036326940522793e-08 | 1.999 | 8.926174771629741e-08 | 1.999 | 1.2623400935662943e-07 | 1.999 |
| 1024×128 | 2.0263647689203173e-08 | 1.988 | 2.2507152948117244e-08 | 1.988 | 3.1829635538382917e-08 | 1.988 |
QUICK
| Mesh | Species concentration L1 error 'ink' | order | Species concentration L2 error 'ink' | order | Species concentration Linf error 'ink' | order |
| 32×4 | 6.3156586487759375e-06 | n/a | 7.000711096885227e-06 | n/a | 9.84847322138549e-06 | n/a |
| 64×8 | 1.363238103921664e-06 | 2.212 | 1.5170194137432052e-06 | 2.206 | 2.145155511279473e-06 | 2.199 |
| 128×16 | 3.266305571818641e-07 | 2.061 | 3.630037585447393e-07 | 2.063 | 5.133586409300506e-07 | 2.063 |
| 256×32 | 8.089394126575096e-08 | 2.014 | 8.984402088331654e-08 | 2.014 | 1.2703997416541046e-07 | 2.015 |
| 512×64 | 2.0382877627860084e-08 | 1.989 | 2.2639206121452135e-08 | 1.989 | 3.201524098606079e-08 | 1.988 |
| 1024×128 | 5.7527909342774115e-09 | 1.825 | 6.389792783510615e-09 | 1.825 | 9.036530912442942e-09 | 1.825 |
HOUC3
| Mesh | Species concentration L1 error 'ink' | order | Species concentration L2 error 'ink' | order | Species concentration Linf error 'ink' | order |
| 32×4 | 3.96202266461209e-06 | n/a | 4.427698046836808e-06 | n/a | 6.257130111375631e-06 | n/a |
| 64×8 | 5.049006637127939e-07 | 2.972 | 5.616713156590353e-07 | 2.979 | 7.941827145918623e-07 | 2.978 |
| 128×16 | 6.571871069880426e-08 | 2.942 | 7.302115148923921e-08 | 2.943 | 1.0326217636524859e-07 | 2.943 |
| 256×32 | 1.0597331361302865e-08 | 2.633 | 1.177099954902994e-08 | 2.633 | 1.6646098721651015e-08 | 2.633 |
| 512×64 | 3.699975782871977e-09 | 1.518 | 4.109573170089247e-09 | 1.518 | 5.8116265133634215e-09 | 1.518 |
| 1024×128 | 2.8375547082295114e-09 | 0.383 | 3.151715418373818e-09 | 0.383 | 4.457159419324341e-09 | 0.383 |
Explicit schemes
o1_upwind
| Mesh | Species concentration L1 error 'ink' | order | Species concentration L2 error 'ink' | order | Species concentration Linf error 'ink' | order |
| 32×4 | 0.00015602765720745208 | n/a | 0.00017370434699546898 | n/a | 0.0002451476395238794 | n/a |
| 64×8 | 7.84023930512313e-05 | 0.993 | 8.713706197308536e-05 | 0.995 | 0.00012316935333056556 | 0.993 |
| 128×16 | 3.9251285385354584e-05 | 0.998 | 4.360488085276386e-05 | 0.999 | 6.165957626602658e-05 | 0.998 |
| 256×32 | 1.963226098979182e-05 | 1.000 | 2.1807154706607713e-05 | 1.000 | 3.083923447255721e-05 | 1.000 |
| 512×64 | 9.81704625971934e-06 | 1.000 | 1.0904208772813554e-05 | 1.000 | 1.5420818003986092e-05 | 1.000 |
| 1024×128 | 4.908659686841856e-06 | 1.000 | 5.452193828066052e-06 | 1.000 | 7.71056405834969e-06 | 1.000 |
o2_upwind
| Mesh | Species concentration L1 error 'ink' | order | Species concentration L2 error 'ink' | order | Species concentration Linf error 'ink' | order |
| 32×4 | 4.0758787536274155e-05 | n/a | 4.519408443172465e-05 | n/a | 6.359941020439575e-05 | n/a |
| 64×8 | 1.02583437266374e-05 | 1.990 | 1.1388894260702232e-05 | 1.989 | 1.6085934853693473e-05 | 1.983 |
| 128×16 | 2.5688415337159686e-06 | 1.998 | 2.852893406757634e-06 | 1.997 | 4.033257891118691e-06 | 1.996 |
| 256×32 | 6.424714489210111e-07 | 1.999 | 7.135780060246731e-07 | 1.999 | 1.0090593465794662e-06 | 1.999 |
| 512×64 | 1.6063355933407326e-07 | 2.000 | 1.7841667053351484e-07 | 2.000 | 2.5231238184354743e-07 | 2.000 |
| 1024×128 | 4.015929351342615e-08 | 2.000 | 4.4605553261033176e-08 | 2.000 | 6.308118889308129e-08 | 2.000 |
HOUC3
| Mesh | Species concentration L1 error 'ink' | order | Species concentration L2 error 'ink' | order | Species concentration Linf error 'ink' | order |
| 32×4 | 3.959295207167299e-06 | n/a | 4.424720095438068e-06 | n/a | 6.252945442319913e-06 | n/a |
| 64×8 | 5.021822453332887e-07 | 2.979 | 5.586655784261863e-07 | 2.986 | 7.899387612919639e-07 | 2.985 |
| 128×16 | 6.300330677766518e-08 | 2.995 | 7.000845622339036e-08 | 2.996 | 9.900327846423806e-08 | 2.996 |
| 256×32 | 7.882728399072358e-09 | 2.999 | 8.75654167555752e-09 | 2.999 | 1.2383537750793039e-08 | 2.999 |
| 512×64 | 9.855758646676966e-10 | 3.000 | 1.0947391783954795e-09 | 3.000 | 1.5481936843642075e-09 | 3.000 |
| 1024×128 | 1.2320476039344707e-10 | 3.000 | 1.3684777294996e-10 | 3.000 | 1.935316351620031e-10 | 3.000 |
WENO3
| Mesh | Species concentration L1 error 'ink' | order | Species concentration L2 error 'ink' | order | Species concentration Linf error 'ink' | order |
| 32×4 | 4.643987882891731e-05 | n/a | 7.935184006431486e-05 | n/a | 0.0002015769258765232 | n/a |
| 64×8 | 7.772605939621686e-06 | 2.579 | 1.7699213896727537e-05 | 2.165 | 6.60715972485848e-05 | 1.609 |
| 128×16 | 8.458199865480319e-07 | 3.200 | 2.352072093041378e-06 | 2.912 | 1.2229192912327491e-05 | 2.434 |
| 256×32 | 9.226216533690971e-08 | 3.197 | 2.460498155802791e-07 | 3.257 | 1.101242707135519e-06 | 3.473 |
| 512×64 | 7.296236433714139e-09 | 3.661 | 2.0920018484498906e-08 | 3.556 | 1.2426408768639874e-07 | 3.148 |
| 1024×128 | 5.279281182109851e-10 | 3.789 | 1.5214235872363999e-09 | 3.781 | 9.445918269790354e-09 | 3.718 |
WENO3_FD
| Mesh | Species concentration L1 error 'ink' | order | Species concentration L2 error 'ink' | order | Species concentration Linf error 'ink' | order |
| 32×4 | 2.108757578366243e-05 | n/a | 2.737431439213543e-05 | n/a | 5.8477573697035545e-05 | n/a |
| 64×8 | 4.20797872686474e-06 | 2.325 | 5.8001813199328735e-06 | 2.239 | 1.52790461145913e-05 | 1.936 |
| 128×16 | 7.455384002289889e-07 | 2.497 | 1.1339586912833923e-06 | 2.355 | 3.848339452328098e-06 | 1.989 |
| 256×32 | 1.2215145877328111e-07 | 2.610 | 2.1152144526826879e-07 | 2.422 | 9.570864408642299e-07 | 2.008 |
| 512×64 | 1.9004167979615335e-08 | 2.684 | 3.8481441057665094e-08 | 2.459 | 2.3526961643455246e-07 | 2.024 |
| 1024×128 | 2.8545782553505617e-09 | 2.735 | 6.9211684980024725e-09 | 2.475 | 5.648033452390622e-08 | 2.058 |
HOUC5
| Mesh | Species concentration L1 error 'ink' | order | Species concentration L2 error 'ink' | order | Species concentration Linf error 'ink' | order |
| 32×4 | 1.205519686220055e-07 | n/a | 1.3497303495190613e-07 | n/a | 1.9081142799137751e-07 | n/a |
| 64×8 | 3.859690529857407e-09 | 4.965 | 4.295806093629425e-09 | 4.974 | 6.074690972468488e-09 | 4.973 |
| 128×16 | 1.213508575310246e-10 | 4.991 | 1.3485932836209207e-10 | 4.993 | 1.9071655366076357e-10 | 4.993 |
| 256×32 | 3.79802965336587e-12 | 4.998 | 4.219175852209917e-12 | 4.998 | 5.966782623545441e-12 | 4.998 |
| 512×64 | 1.1875779358386587e-13 | 4.999 | 1.3188844384521593e-13 | 5.000 | 1.8685053504441385e-13 | 4.997 |
| 1024×128 | 3.7480064421524e-15 | 4.986 | 4.1614965128099156e-15 | 4.986 | 6.439293542825908e-15 | 4.859 |
WENO5
| Mesh | Species concentration L1 error 'ink' | order | Species concentration L2 error 'ink' | order | Species concentration Linf error 'ink' | order |
| 32×4 | 2.818007842110559e-07 | n/a | 3.590608745620612e-07 | n/a | 5.656440379420147e-07 | n/a |
| 64×8 | 1.037636756661027e-08 | 4.763 | 1.1521856482173458e-08 | 4.962 | 2.2397949717145593e-08 | 4.658 |
| 128×16 | 2.9985008020375764e-10 | 5.113 | 3.42829009073388e-10 | 5.071 | 8.974173626441484e-10 | 4.641 |
| 256×32 | 8.604534904141543e-12 | 5.123 | 1.0039503310523777e-11 | 5.094 | 3.1779134879172943e-11 | 4.820 |
| 512×64 | 2.539427374707425e-13 | 5.083 | 2.935784785563632e-13 | 5.096 | 1.0460521338018225e-12 | 4.925 |
| 1024×128 | 7.65325862173193e-15 | 5.052 | 8.720614047945479e-15 | 5.073 | 3.297362383136715e-14 | 4.987 |
WENO5_FD
| Mesh | Species concentration L1 error 'ink' | order | Species concentration L2 error 'ink' | order | Species concentration Linf error 'ink' | order |
| 32×4 | 1.2780101635977842e-06 | n/a | 1.402047505866947e-06 | n/a | 1.7960821623308476e-06 | n/a |
| 64×8 | 3.4031184616405835e-08 | 5.231 | 3.817012818446319e-08 | 5.199 | 5.8944604419686186e-08 | 4.929 |
| 128×16 | 9.568043511350502e-10 | 5.152 | 1.0830740208280564e-09 | 5.139 | 1.8262753531672615e-09 | 5.012 |
| 256×32 | 2.8236688771128733e-11 | 5.083 | 3.186641896760519e-11 | 5.087 | 5.691258575524216e-11 | 5.004 |
| 512×64 | 8.566006350150656e-13 | 5.043 | 9.611694179796472e-13 | 5.051 | 1.7783552408445757e-12 | 5.000 |
| 1024×128 | 2.6377665615715928e-14 | 5.021 | 2.9452510612823703e-14 | 5.028 | 5.5844218138645374e-14 | 4.993 |
References
1.11.0