An Unconditionally Stable Numerical Scheme for 3D Coupled Burgers’ Equations
Abstract
:1. Introduction
2. Materials and Methods
2.1. Fractional Step Method
- Fractional step decomposition—The three-dimensional coupled Burgers’ equations are efficiently transformed into simpler one-dimensional forms, reducing computational effort;
- Implicit finite difference scheme—Unlike explicit methods, the implicit approach enhances numerical stability and accuracy;
- Unconditional stability—A rigorous von Neumann stability analysis proves that the method is unconditionally stable, ensuring reliable numerical solutions;
- Benchmark validation—The method is tested on standard benchmark problems, demonstrating superior accuracy and robustness compared to existing techniques;
- Computational efficiency—The decomposition strategy reduces computational costs while maintaining high accuracy, making the approach suitable for large-scale simulations in fluid dynamics and related fields.
2.2. Local Truncation Error (LTE) and Consistency
2.3. Stability Analysis
2.4. Convergence
3. Results
4. Conclusions
5. Note
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Exact | Present | Absolute Error | ||
---|---|---|---|---|
0.002 | (0.1, 0.1, 0.1) (0.1, 0.1, 0.5) (0.1, 0.1, 0.9) (0.5, 0.1, 0.1) (0.5, 0.1, 0.5) (0.5, 0.1, 0.9) (0.5, 0.5, 0.5) (0.5, 0.5, 0.9) (0.9, 0.1, 0.1) (0.9, 0.1, 0.5) (0.9, 0.5, 0.5) (0.9, 0.9, 0.5) (0.9, 0.9, 0.9) | −0.01834520 −0.01896376 −0.01945593 −0.01340710 −0.01368335 −0.01389983 −0.01492331 −0.01582336 −0.01054814 −0.01062954 −0.01098810 −0.01124165 −0.01160368 | −0.01834552 −0.01896527 −0.01945839 −0.01340724 −0.01368402 −0.01390090 −0.01492619 −0.01582768 −0.01054819 −0.01062973 −0.01098892 −0.01124285 −0.01160535 | |
0.006 | (0.1, 0.1, 0.1) (0.1, 0.1, 0.5) (0.1, 0.1, 0.9) (0.5, 0.1, 0.1) (0.5, 0.1, 0.5) (0.5, 0.1, 0.9) (0.5, 0.5, 0.5) (0.5, 0.5, 0.9) (0.9, 0.1, 0.1) (0.9, 0.1, 0.5) (0.9, 0.5, 0.5) (0.9, 0.9, 0.5) (0.9, 0.9, 0.9) | −0.01834455 −0.01896065 −0.01945088 −0.01340680 −0.01368198 −0.01389762 −0.01491740 −0.01581448 −0.01054806 −0.01062913 −0.01098642 −0.01123917 −0.01160025 | −0.01834550 −0.01896518 −0.01945824 −0.01340723 −0.01368398 −0.01390084 −0.01492601 −0.01582741 −0.01054818 −0.01062972 −0.01098887 −0.01124278 −0.01160525 | |
0.010 | (0.1, 0.1, 0.1) (0.1, 0.1, 0.5) (0.1, 0.1, 0.9) (0.5, 0.1, 0.1) (0.5, 0.1, 0.5) (0.5, 0.1, 0.9) (0.5, 0.5, 0.5) (0.5, 0.5, 0.9) (0.9, 0.1, 0.1) (0.9, 0.1, 0.5) (0.9, 0.5, 0.5) (0.9, 0.9, 0.5) (0.9, 0.9, 0.9) | −0.01834390 −0.01895756 −0.01944585 −0.01340651 −0.01368061 −0.01389542 −0.01491150 −0.01580563 −0.01054797 −0.01062873 −0.01098474 −0.01123671 −0.01159682 | −0.01834548 −0.01896508 −0.01945809 −0.01340722 −0.01368394 −0.01390077 −0.01492583 −0.01582715 −0.01054818 −0.01062971 −0.01098882 −0.01124270 −0.01160515 |
Exact | Present | Absolute Error | ||
---|---|---|---|---|
0.002 | (0.1, 0.1, 0.1) (0.1, 0.1, 0.5) (0.1, 0.1, 0.9) (0.5, 0.1, 0.1) (0.5, 0.1, 0.5) (0.5, 0.1, 0.9) (0.5, 0.5, 0.5) (0.5, 0.5, 0.9) (0.9, 0.1, 0.1) (0.9, 0.1, 0.5) (0.9, 0.5, 0.5) (0.9, 0.9, 0.5) (0.9, 0.9, 0.9) | −0.00017979 −0.00086042 −0.00140200 −0.00063170 −0.00299749 −0.00485131 −0.00250076 −0.00391638 −0.00081409 −0.00384970 −0.00316312 −0.00212417 −0.00319923 | −0.00018013 −0.00086209 −0.00140470 −0.00063293 −0.00300322 −0.00486050 −0.00250529 −0.00392317 −0.00081567 −0.00385704 −0.00316873 −0.00212774 −0.00320419 | |
0.006 | (0.1, 0.1, 0.1) (0.1, 0.1, 0.5) (0.1, 0.1, 0.9) (0.5, 0.1, 0.1) (0.5, 0.1, 0.5) (0.5, 0.1, 0.9) (0.5, 0.5, 0.5) (0.5, 0.5, 0.9) (0.9, 0.1, 0.1) (0.9, 0.1, 0.5) (0.9, 0.5, 0.5) (0.9, 0.9, 0.5) (0.9, 0.9, 0.9) | −0.00017907 −0.00085700 −0.00139644 −0.00062919 −0.00298570 −0.00483241 −0.00249146 −0.00390242 −0.00081086 −0.00383463 −0.00315158 −0.00211682 −0.00318902 | −0.00018011 −0.00086198 −0.00140453 −0.00063285 −0.00300286 −0.00485993 −0.00250501 −0.00392275 −0.00081557 −0.00385658 −0.00316838 −0.00212752 −0.00320388 | |
0.010 | (0.1, 0.1, 0.1) (0.1, 0.1, 0.5) (0.1, 0.1, 0.9) (0.5, 0.1, 0.1) (0.5, 0.1, 0.5) (0.5, 0.1, 0.9) (0.5, 0.5, 0.5) (0.5, 0.5, 0.9) (0.9, 0.1, 0.1) (0.9, 0.1, 0.5) (0.9, 0.5, 0.5) (0.9, 0.9, 0.5) (0.9, 0.9, 0.9) | −0.00017835 −0.00085360 −0.00139090 −0.00062669 −0.00297396 −0.00481359 −0.00248219 −0.00388849 −0.00080763 −0.00381962 −0.00314008 −0.00210949 −0.00317884 | −0.00018009 −0.00086188 −0.00140437 −0.00063278 −0.00300251 −0.00485936 −0.00250473 −0.00392233 −0.00081547 −0.00385613 −0.00316804 −0.00212730 −0.00320357 |
Exact | Present | Absolute Error | ||
---|---|---|---|---|
0.002 | (0.1, 0.1, 0.1) (0.1, 0.1, 0.5) (0.1, 0.1, 0.9) (0.5, 0.1, 0.1) (0.5, 0.1, 0.5) (0.5, 0.1, 0.9) (0.5, 0.5, 0.5) (0.5, 0.5, 0.9) (0.9, 0.1, 0.1) (0.9, 0.1, 0.5) (0.9, 0.5, 0.5) (0.9, 0.9, 0.5) (0.9, 0.9, 0.9) | −0.00017979 −0.00015803 −0.00011163 −0.00063170 −0.00055052 −0.00038626 −0.00250076 −0.00169783 −0.00081409 −0.00070704 −0.00316312 −0.00489982 −0.00319923 | −0.00018013 −0.00015833 −0.00011184 −0.00063293 −0.00055158 −0.00038700 −0.00250529 −0.00170077 −0.00081567 −0.00070839 −0.00316873 −0.00490806 −0.00320419 | |
0.006 | (0.1, 0.1, 0.1) (0.1, 0.1, 0.5) (0.1, 0.1, 0.9) (0.5, 0.1, 0.1) (0.5, 0.1, 0.5) (0.5, 0.1, 0.9) (0.5, 0.5, 0.5) (0.5, 0.5, 0.9) (0.9, 0.1, 0.1) (0.9, 0.1, 0.5) (0.9, 0.5, 0.5) (0.9, 0.9, 0.5) (0.9, 0.9, 0.9) | −0.00017907 −0.00015740 −0.00011119 −0.00062919 −0.00054836 −0.00038476 −0.00249146 −0.00169177 −0.00081086 −0.00070427 −0.00315158 −0.00488287 −0.00318902 | −0.00018011 −0.00015831 −0.00011183 −0.00063285 −0.00055151 −0.00038695 −0.00250501 −0.00170059 −0.00081557 −0.00070831 −0.00316838 −0.00490755 −0.00320388 | |
0.010 | (0.1, 0.1, 0.1) (0.1, 0.1, 0.5) (0.1, 0.1, 0.9) (0.5, 0.1, 0.1) (0.5, 0.1, 0.5) (0.5, 0.1, 0.9) (0.5, 0.5, 0.5) (0.5, 0.5, 0.9) (0.9, 0.1, 0.1) (0.9, 0.1, 0.5) (0.9, 0.5, 0.5) (0.9, 0.9, 0.5) (0.9, 0.9, 0.9) | −0.00017835 −0.00015677 −0.00011074 −0.00062669 −0.00054620 −0.00038326 −0.00248219 −0.00168574 −0.00080763 −0.00070152 −0.00314008 −0.00486597 −0.00317884 | −0.00018009 −0.00015829 −0.00011182 −0.00063278 −0.00055144 −0.00038691 −0.00250473 −0.00170041 −0.00081547 −0.00070822 −0.00316804 −0.00490704 −0.00320357 |
Present | LT-ADM [25] | LT-HPM [25] | VIM [25] | VIDM [25] | VIHPM [25] | CPU | |
---|---|---|---|---|---|---|---|
0.002 | 47 | ||||||
0.004 | 78 | ||||||
0.006 | 105 | ||||||
0.008 | 156 | ||||||
0.010 | 165 |
Present | LT-ADM [25] | LT-HPM [25] | VIM [25] | VIDM [25] | VIHPM [25] | |
---|---|---|---|---|---|---|
0.002 | ||||||
0.004 | ||||||
0.006 | ||||||
0.008 | ||||||
0.010 |
Present | LT-ADM [25] | LT-HPM [25] | VIM [25] | VIDM [25] | VIHPM [25] | |
---|---|---|---|---|---|---|
0.002 | ||||||
0.004 | ||||||
0.006 | ||||||
0.008 | ||||||
0.010 |
CPU | |||||||
---|---|---|---|---|---|---|---|
0.1 | 0.01607602 | 0.01701311 | 0.01701314 | 0.00078824 | 0.00052727 | 0.00052727 | 18 |
0.2 | 0.02885929 | 0.03118231 | 0.03118236 | 0.00136220 | 0.00094535 | 0.00094535 | 36 |
0.3 | 0.03925681 | 0.04314278 | 0.04314281 | 0.00180512 | 0.00129970 | 0.00129970 | 54 |
0.4 | 0.04777030 | 0.05327088 | 0.05327092 | 0.00211052 | 0.00158540 | 0.00158540 | 72 |
0.5 | 0.05473662 | 0.06183566 | 0.06183568 | 0.00234148 | 0.00182888 | 0.00182888 | 89 |
0.6 | 0.06040639 | 0.06904788 | 0.06904791 | 0.00256857 | 0.00203571 | 0.00203571 | 107 |
0.7 | 0.06497688 | 0.07508099 | 0.07508107 | 0.00273030 | 0.00220845 | 0.00220846 | 125 |
0.8 | 0.06860875 | 0.08008187 | 0.08008195 | 0.00283681 | 0.00234598 | 0.00234599 | 143 |
0.9 | 0.07144140 | 0.08418134 | 0.08418139 | 0.00289775 | 0.00245845 | 0.00245846 | 161 |
1.0 | 0.07357237 | 0.08747612 | 0.08747618 | 0.00292069 | 0.00254495 | 0.00254495 | 178 |
CPU | |||||||
---|---|---|---|---|---|---|---|
10 | 0.12595460 | 0.17624530 | 0.17628540 | 0.00398876 | 0.00496712 | 0.00496995 | |
20 | 0.14303290 | 0.19327000 | 0.19330240 | 0.00473712 | 0.00550840 | 0.00551130 | |
30 | 0.13336220 | 0.17461560 | 0.17464180 | 0.00461626 | 0.00496580 | 0.00496805 | |
40 | 0.12094910 | 0.15458880 | 0.15460980 | 0.00431160 | 0.00443582 | 0.00443767 | |
50 | 0.10965810 | 0.13755450 | 0.13757130 | 0.00401893 | 0.00393986 | 0.00394031 | 2 |
60 | 0.09998650 | 0.12355520 | 0.12356920 | 0.00377040 | 0.00356142 | 0.00356287 | |
70 | 0.09178136 | 0.11202780 | 0.11203940 | 0.00352644 | 0.00324337 | 0.00324450 | |
80 | 0.08479263 | 0.10243280 | 0.10244300 | 0.00329654 | 0.00296713 | 0.00296735 | |
90 | 0.07879200 | 0.09434658 | 0.09435534 | 0.00308419 | 0.00273646 | 0.00273663 | |
100 | 0.07359336 | 0.08744954 | 0.08745717 | 0.00291878 | 0.00254064 | 0.00254077 | |
1000 | 0.01124273 | 0.01200673 | 0.01200697 | 0.00054696 | 0.00036272 | 0.00036273 | |
10000 | 0.00122298 | 0.00127331 | 0.00127331 | 0.00006759 | 0.00003947 | 0.00003947 | 3 |
Grid Point | ||||||
---|---|---|---|---|---|---|
(0.1, 0.1, 0.9) | 0.0239584 −0.1433502 −0.3098060 | 0.0251936 −0.0942885 −0.0444397 | 0.0198349 −0.0019673 0.0571594 | 0.0087784 0.0161314 0.0209709 | 0.0025408 0.0079498 0.0053030 | |
(0.1, 0.1, 0.1) | 0.0130397 0.2961703 0.5933083 | 0.0086755 0.1819000 0.2619675 | 0.0042517 0.0713250 0.0521275 | 0.0021648 0.0258374 0.0125001 | 0.0010801 0.0071994 0.0034688 | |
(0.9, 0.1, 0.1) | 0.0373669 −0.2133582 −0.3512228 | 0.0350676 −0.1539166 −0.0849666 | 0.0219688 −0.0399523 0.0298473 | 0.0091082 0.0024782 0.0171387 | 0.0024632 0.0052484 0.0049001 | |
(0.9, 0.1, 0.9) | 0.0287756 0.0465253 0.1132805 | 0.0261095 0.0281507 0.0222440 | 0.0290998 0.0072191 0.0443088 | 0.0224096 0.0118066 0.0329812 | 0.0056462 0.0084688 0.0085465 | |
(0.5, 0.5, 0.5) | 0.8804632 1.7752066 0.2335210 | 0.6788833 2.0130337 0.1981508 | 0.3317227 1.3315012 0.3736063 | 0.1704326 0.6964189 0.3278693 | 0.0806389 0.2940547 0.1650422 | |
(0.1, 0.9, 0.9) | 0.0293427 0.0473384 0.1131651 | 0.0258647 0.0287051 0.0199498 | 0.0209887 0.0078080 0.0268328 | 0.0157792 0.0255877 0.0232304 | 0.0063097 0.0232738 0.0108606 | |
(0.1, 0.9, 0.1) | 0.0285317 −0.1743388 −0.3571369 | 0.0222670 −0.1143676 −0.1186861 | 0.0122100 −0.0169228 0.0035327 | 0.0080087 0.0576209 0.0156805 | 0.0027520 0.0189870 0.0066534 | |
(0.9, 0.9, 0.1) | 0.0274635 0.0434720 0.1061607 | 0.0280131 0.0281587 0.0114414 | 0.0204932 0.0042559 0.0088801 | 0.0117795 0.0117706 0.0139944 | 0.0049298 0.0145751 0.0080140 | |
(0.9, 0.9, 0.9) | 0.0271956 −0.0122070 −0.0415867 | 0.0264742 −0.0067082 −0.0019528 | 0.0306129 0.0000573 0.0280496 | 0.0370216 0.0201885 0.0365225 | 0.0153747 0.0316911 0.0195732 | |
CPU | 4 | 8 | 19 | 38 | 75 |
Grid Point | Method | |||||
---|---|---|---|---|---|---|
(0.1, 0.1, 0.9) | Present DQ-FDM [24] FDM [24] | 0.0254111 0.02364 0.025403 | 0.0115768 0.01076 0.011756 | 0.0041839 0.00391 0.004266 | 0.0009462 0.00090 0.000983 | |
Present DQ-FDM [24] FDM [24] | −0.1071553 −0.10201 −0.108493 | 0.0109503 0.00971 0.010697 | 0.0093759 0.00872 0.009521 | 0.0024052 0.00227 0.002498 | ||
Present DQ-FDM [24] FDM [24] | −0.0680342 −0.07772 −0.089963 | 0.0185890 0.01747 0.020117 | 0.0058328 0.00545 0.005872 | 0.0012179 0.00115 0.001228 | ||
(0.1, 0.1, 0.1) | Present DQ-FDM [24] FDM [24] | 0.0135700 0.01291 0.013941 | 0.0043702 0.00413 0.004521 | 0.0022730 0.00214 0.002331 | 0.0007671 0.00073 0.000798 | |
Present DQ-FDM [24] FDM [24] | 0.2722870 0.26139 0.284202 | 0.0456060 0.04368 0.048848 | 0.0124520 0.01180 0.013223 | 0.0023299 0.00221 0.002449 | ||
Present DQ-FDM [24] FDM [24] | 0.3243355 0.32746 0.379733 | 0.0148860 0.01391 0.016177 | 0.0042409 0.00391 0.004263 | 0.0010379 0.00097 0.001045 | ||
(0.5, 0.5, 0.5) | Present DQ-FDM [24] FDM [24] | 0.8221614 0.82203 0.826809 | 0.3196220 0.32429 0.328282 | 0.1489025 0.15109 0.152905 | 0.0361105 0.03715 0.037998 | |
Present DQ-FDM [24] FDM [24] | 1.5623827 1.57094 1.540632 | 0.9474787 0.94780 0.971748 | 0.4004099 0.40393 0.409879 | 0.0912639 0.09344 0.094790 | ||
Present DQ-FDM [24] FDM [24] | 0.3610981 0.37251 0.401473 | 0.3579388 0.35500 0.340496 | 0.1880154 0.18819 0.185145 | 0.0451686 0.04595 0.045831 | ||
(0.9, 0.9, 0.1) | Present DQ-FDM [24] FDM [24] | 0.0266505 0.02445 0.026479 | 0.0129122 0.01202 0.013229 | 0.0063366 0.00592 0.006476 | 0.0012449 0.00118 0.001334 | |
Present DQ-FDM [24] FDM [24] | 0.0279643 0.02662 0.029084 | 0.0038226 0.00378 0.003146 | 0.0113319 0.01041 0.010327 | 0.0030525 0.00288 0.003158 | ||
Present DQ-FDM [24] FDM [24] | 0.0139741 0.02232 0.032489 | 0.0133409 0.01219 0.012178 | 0.0068297 0.00626 0.006790 | 0.0015038 0.00141 0.001544 | ||
(0.9, 0.9, 0.9) | Present DQ-FDM [24] FDM [24] | 0.0259294 0.02393 0.025940 | 0.0206922 0.01909 0.020358 | 0.0103278 0.00958 0.010436 | 0.0015613 0.00149 0.001669 | |
Present DQ-FDM [24] FDM [24] | −0.0067402 −0.00660 −0.007179 | 0.0023861 0.00250 0.002243 | 0.0104624 0.00969 0.009546 | 0.0032999 0.00311 0.003387 | ||
Present DQ-FDM [24] FDM [24] | −0.0027490 −0.00627 −0.006416 | 0.0218266 0.02055 0.020134 | 0.0103696 0.00968 0.010395 | 0.0018227 0.00172 0.001875 | ||
CPU | 3 | 19 | 38 | 75 |
Grid Point | Method | |||||
---|---|---|---|---|---|---|
(0.1, 0.1, 0.9) | Present DQ-FDM [24] FDM [24] | 0.0207517 0.01919 0.020817 | 0.0038962 0.00370 0.004033 | 0.0006396 0.00062 0.000682 | 0.0000164 0.00002 0.000019 | |
Present DQ-FDM [24] FDM [24] | −0.0310491 −0.03350 −0.035715 | 0.0066557 0.00622 0.006890 | 0.0010418 0.00100 0.001105 | 0.0000260 0.00003 0.000030 | ||
Present DQ-FDM [24] FDM [24] | 0.0279222 0.02000 0.026177 | 0.0042174 0.00401 0.004361 | 0.0006661 0.00064 0.000706 | 0.0000170 0.00002 0.000020 | ||
(0.1, 0.1, 0.1) | Present DQ-FDM [24] FDM [24] | 0.0131351 0.01247 0.013573 | 0.0031357 0.00297 0.003242 | 0.0006058 0.00058 0.000645 | 0.0000164 0.00002 0.000019 | |
Present DQ-FDM [24] FDM [24] | 0.1831330 0.17987 0.197984 | 0.0099416 0.00978 0.010907 | 0.0010810 0.00105 0.001159 | 0.0000260 0.00003 0.000030 | ||
Present DQ-FDM [24] FDM [24] | 0.0690051 0.07888 0.095820 | 0.0035683 0.00339 0.003684 | 0.0006332 0.00061 0.000746 | 0.0000169 0.00002 0.000020 | ||
(0.5, 0.5, 0.5) | Present DQ-FDM [24] FDM [24] | 0.6654563 0.66955 0.672757 | 0.1411316 0.14484 0.146526 | 0.0223667 0.02345 0.023959 | 0.0005572 0.00063 0.000654 | |
Present DQ-FDM [24] FDM [24] | 1.0757338 1.06574 1.055410 | 0.2289045 0.23342 0.233923 | 0.0354140 0.03684 0.037139 | 0.0008814 0.00098 0.001013 | ||
Present DQ-FDM [24] FDM [24] | 0.5259060 0.53003 0.531746 | 0.1456256 0.14901 0.149800 | 0.0231289 0.02420 0.024597 | 0.0005761 0.00065 0.000672 | ||
(0.9, 0.9, 0.1) | Present DQ-FDM [24] FDM [24] | 0.0205042 0.01899 0.020760 | 0.0045598 0.00431 0.004726 | 0.0006817 0.00066 0.000733 | 0.0000165 0.00002 0.000019 | |
Present DQ-FDM [24] FDM [24] | 0.0047205 0.00592 0.006935 | 0.0051581 0.00468 0.004731 | 0.0010428 0.00100 0.001076 | 0.0000260 0.00003 0.000030 | ||
Present DQ-FDM [24] FDM [24] | 0.0080648 0.00749 0.0053180 | 0.0045400 0.00428 0.004689 | 0.0006999 0.00068 0.000747 | 0.0000170 0.00002 0.000020 | ||
(0.9, 0.9, 0.9) | Present DQ-FDM [24] FDM [24] | 0.0212308 0.01971 0.021345 | 0.0053239 0.00502 0.005498 | 0.0007187 0.00070 0.000775 | 0.0000165 0.00002 0.000019 | |
Present DQ-FDM [24] FDM [24] | −0.0006289 −0.00100 −0.001143 | 0.0035969 0.00323 0.003274 | 0.0010081 0.00096 0.001032 | 0.0000260 0.00003 0.000030 | ||
Present DQ-FDM [24] FDM [24] | 0.0051624 0.00529 0.004513 | 0.0053273 0.00503 0.005506 | 0.0007364 0.00071 0.000788 | 0.0000171 0.00002 0.000020 | ||
CPU | 4 | 19 | 37 | 75 |
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Çelikten, G. An Unconditionally Stable Numerical Scheme for 3D Coupled Burgers’ Equations. Symmetry 2025, 17, 452. https://doi.org/10.3390/sym17030452
Çelikten G. An Unconditionally Stable Numerical Scheme for 3D Coupled Burgers’ Equations. Symmetry. 2025; 17(3):452. https://doi.org/10.3390/sym17030452
Chicago/Turabian StyleÇelikten, Gonca. 2025. "An Unconditionally Stable Numerical Scheme for 3D Coupled Burgers’ Equations" Symmetry 17, no. 3: 452. https://doi.org/10.3390/sym17030452
APA StyleÇelikten, G. (2025). An Unconditionally Stable Numerical Scheme for 3D Coupled Burgers’ Equations. Symmetry, 17(3), 452. https://doi.org/10.3390/sym17030452