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version 0.6.0
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version 0.6.0
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This page demonstrates the temporal convergence rates of diffusion schemes.
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
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.
Conjointly with temporal schemes, spatial schemes are also tested in order to test all possible configurations.
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.
All schemes exhibit the expected convergence rate.
All schemes exhibit the expected convergence rate.
| Time step | Temperature L1 error | order | Temperature L2 error | order | Temperature Linf error | order |
|---|---|---|---|---|---|---|
| 0.0004 | 9.276729804807057e-06 | n/a | 9.839457828310192e-06 | n/a | 1.3915094707162012e-05 | n/a |
| 0.0002 | 9.275762316305705e-06 | 0.000 | 9.838431651804874e-06 | 0.000 | 1.3913643474472437e-05 | 0.000 |
| 0.0001 | 9.275071235756438e-06 | 0.000 | 9.837698650172784e-06 | 0.000 | 1.3912606853572207e-05 | 0.000 |
| 5e-05 | 9.274725690427976e-06 | 0.000 | 9.83733214399062e-06 | 0.000 | 1.3912088535517064e-05 | 0.000 |
| 2.5e-05 | 9.274552916267258e-06 | 0.000 | 9.837148889291828e-06 | 0.000 | 1.3911829374158025e-05 | 0.000 |
| 1.25e-05 | 9.274466524456884e-06 | 0.000 | 9.837057257027716e-06 | 0.000 | 1.3911699786817167e-05 | 0.000 |
| Time step | Temperature L1 error | order | Temperature L2 error | order | Temperature Linf error | order |
|---|---|---|---|---|---|---|
| 0.0004 | 4.2360540489451017e-07 | n/a | 4.695207420764847e-07 | n/a | 6.598274921998382e-07 | n/a |
| 0.0002 | 2.1220687993364432e-07 | 0.997 | 2.3573364842150676e-07 | 0.994 | 3.333118240300692e-07 | 0.985 |
| 0.0001 | 1.0557679916893944e-07 | 1.007 | 1.1720483959890876e-07 | 1.008 | 1.654919158378121e-07 | 1.010 |
| 5e-05 | 5.2996894370210315e-08 | 0.994 | 5.887250813916643e-08 | 0.993 | 8.324184197494588e-08 | 0.991 |
| 2.5e-05 | 2.636984415244257e-08 | 1.007 | 2.9285691958234012e-08 | 1.007 | 4.139993214913318e-08 | 1.008 |
| 1.25e-05 | 1.3163942103967027e-08 | 1.002 | 1.4621583124199109e-08 | 1.002 | 2.0677796142010152e-08 | 1.002 |
| Time step | Temperature L1 error | order | Temperature L2 error | order | Temperature Linf error | order |
|---|---|---|---|---|---|---|
| 0.0004 | 2.2747162879979684e-08 | n/a | 2.5267075483584256e-08 | n/a | 3.573021956082556e-08 | n/a |
| 0.0002 | 5.700560175354866e-09 | 1.997 | 6.331563737938864e-09 | 1.997 | 8.954183061682386e-09 | 1.997 |
| 0.0001 | 1.4241289490908561e-09 | 2.001 | 1.5817991487475524e-09 | 2.001 | 2.236958174250958e-09 | 2.001 |
| 5e-05 | 3.560366988253703e-10 | 2.000 | 3.954567053705981e-10 | 2.000 | 5.592584173541582e-10 | 2.000 |
| 2.5e-05 | 8.901436694290283e-11 | 2.000 | 9.887006409680226e-11 | 2.000 | 1.3982381918964393e-10 | 2.000 |
| 1.25e-05 | 2.226099009763731e-11 | 2.000 | 2.4725408647574382e-11 | 2.000 | 3.4972691409507206e-11 | 1.999 |
| Time step | Temperature L1 error | order | Temperature L2 error | order | Temperature Linf error | order |
|---|---|---|---|---|---|---|
| 0.0004 | 9.272030321800611e-06 | n/a | 9.834473273876924e-06 | n/a | 1.390804548273561e-05 | n/a |
| 0.0002 | 9.272997914117065e-06 | -0.000 | 9.835499560504601e-06 | -0.000 | 1.3909496871189475e-05 | -0.000 |
| 0.0001 | 9.273689034537218e-06 | -0.000 | 9.83623260440816e-06 | -0.000 | 1.3910533551819704e-05 | -0.000 |
| 5e-05 | 9.274034589635643e-06 | -0.000 | 9.836599120926768e-06 | -0.000 | 1.3911051884418768e-05 | -0.000 |
| 2.5e-05 | 9.274207366224974e-06 | -0.000 | 9.836782378180229e-06 | -0.000 | 1.3911311049330521e-05 | -0.000 |
| 1.25e-05 | 9.274293754399365e-06 | -0.000 | 9.83687400666958e-06 | -0.000 | 1.3911440631564354e-05 | -0.000 |
| Time step | Temperature L1 error | order | Temperature L2 error | order | Temperature Linf error | order |
|---|---|---|---|---|---|---|
| 0.0004 | 1.2728223023015612e-06 | n/a | 1.3500319218027258e-06 | n/a | 1.9092334534454025e-06 | n/a |
| 0.0002 | 1.2739578709129267e-06 | -0.001 | 1.3512363741978537e-06 | -0.001 | 1.9109368063485732e-06 | -0.001 |
| 0.0001 | 1.2747689694438586e-06 | -0.001 | 1.3520966741017311e-06 | -0.001 | 1.9121534541310936e-06 | -0.001 |
| 5e-05 | 1.2751745127881353e-06 | -0.000 | 1.3525268177863845e-06 | -0.000 | 1.9127617691960808e-06 | -0.000 |
| 2.5e-05 | 1.2753772829175265e-06 | -0.000 | 1.3527418879752074e-06 | -0.000 | 1.913065924341595e-06 | -0.000 |
| 1.25e-05 | 1.2754786678087493e-06 | -0.000 | 1.3528494228946192e-06 | -0.000 | 1.9132180016923073e-06 | -0.000 |
| Time step | Temperature L1 error | order | Temperature L2 error | order | Temperature Linf error | order |
|---|---|---|---|---|---|---|
| 0.0004 | 4.219651260707811e-07 | n/a | 4.67702670532917e-07 | n/a | 6.572725175235661e-07 | n/a |
| 0.0002 | 2.1173475717125858e-07 | 0.995 | 2.3520918276350523e-07 | 0.992 | 3.3257026399446943e-07 | 0.983 |
| 0.0001 | 1.054579553492955e-07 | 1.006 | 1.1707290652640336e-07 | 1.007 | 1.653056258543728e-07 | 1.009 |
| 5e-05 | 5.296717294080344e-08 | 0.993 | 5.8839491582548983e-08 | 0.993 | 8.319515854005033e-08 | 0.991 |
| 2.5e-05 | 2.6362410617225384e-08 | 1.007 | 2.9277436497879112e-08 | 1.007 | 4.1388260485497597e-08 | 1.007 |
| 1.25e-05 | 1.316209225829936e-08 | 1.002 | 1.4619528444039825e-08 | 1.002 | 2.0674881140436696e-08 | 1.001 |
| Time step | Temperature L1 error | order | Temperature L2 error | order | Temperature Linf error | order |
|---|---|---|---|---|---|---|
| 0.0004 | 5.939646256996798e-08 | n/a | 6.629583689209785e-08 | n/a | 9.307288140192327e-08 | n/a |
| 0.0002 | 2.9692044959825776e-08 | 1.000 | 3.3007824407029854e-08 | 1.006 | 4.662019936763073e-08 | 0.997 |
| 0.0001 | 1.4699028720955808e-08 | 1.014 | 1.6336331684022464e-08 | 1.015 | 2.3082245004246715e-08 | 1.014 |
| 5e-05 | 7.401713112369977e-09 | 0.990 | 8.193323829015775e-09 | 0.996 | 1.1587109383093264e-08 | 0.994 |
| 2.5e-05 | 3.510895851430677e-09 | 1.076 | 3.900767008526426e-09 | 1.071 | 5.514094514325052e-09 | 1.071 |
| 1.25e-05 | 1.7612999763920988e-09 | 0.995 | 1.956717899740599e-09 | 0.995 | 2.7663551449563784e-09 | 0.995 |
1.11.0