Numerical Analysis of Convection–Diffusion Using a Modified Upwind Approach in the Finite Volume Method
Abstract
:1. Introduction
2. Preliminary Remarks
3. Discretization and Formulation of the Problem
4. The Formulation of the Problem within the Analysis of Boundary Conditions
5. Theoretical Aspects and Analysis of the Numerical Scheme
5.1. Stability Analysis
5.2. Convergence Analysis of the Scheme
6. Numerical Experiments
7. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Max-Error | Order | Max-Error | Order | Max-Error | Order | ||
1/10 | 0.0089 | – | 0.0089 | – | 0.0089 | – | |
1/20 | 0.0025 | 1.8149 | 0.0025 | 1.8145 | 0.0025 | 1.8144 | |
1/40 | 6.8518 × | 1.8886 | 6.8620 × | 1.8871 | 6.86404 × | 1.8868 | |
1/80 | 1.7773 × | 1.9468 | 1.7867 × | 1.9414 | 1.8784 × | 1.9404 |
1/300 | 1/1500 | ||||||
---|---|---|---|---|---|---|---|
Max-Error | Order | Max-Error | Order | Max-Error | Order | ||
1/20 | 0.0022 | – | 0.0022 | – | 0.0022 | – | |
1/40 | 5.4741 × | 2.0029 | 5.5206 × | 2.0071 | 6.80408 × | 2.0019 | |
1/80 | 1.3325 × | 2.0060 | 1.3744 × | 2.0227 | 1.7884 × | 2.0019 | |
1/160 | 3.07300 × | 2.0025 | 3.3829 × | 2.0789 | 4.5566 × | 2.0065 |
10,000 | ||||||||
---|---|---|---|---|---|---|---|---|
FVM (13) | FVM [21] | FVM (13) | FVM [21] | |||||
Max-Error | Order | Max-Error | Order | Max-Error | Order | Max-Error | Order | |
1/10 | 0.0038 | – | 0.0154 | – | 0.0038 | – | 0.0154 | – |
1/20 | 0.0012 | 1.7261 | 0.0081 | 0.9327 | 0.0012 | 1.7261 | 0.0083 | 0.9312 |
1/40 | 3.2596 | 1.8467 | 0.0041 | 0.9644 | 3.2538 | 1.8467 | 0.0042 | 0.9632 |
1/80 | 8.7216 | 1.9055 | 0.0021 | 0.9924 | 8.7426 | 1.9048 | 0.0021 | 0.98151 |
FVM (13) | FVM [21] | FVM (13) | FVM [21] | |||||
Max-Error | Order | Max-Error | Order | Max-Error | Order | Max-Error | Order | |
1/20 | 7.6965 | – | 0.0107 | – | 7.7013 | – | 0.0108 | – |
1/40 | 1.8748 | 2.0375 | 0.0054 | 0.9927 | 1.8793 | 2.0349 | 0.0055 | 0.9712 |
1/80 | 4.6073 | 2.0248 | 0.0025 | 1.0953 | 4.6437 | 2.0169 | 0.0026 | 1.0572 |
1/160 | 1.2930 | 1.8332 | 0.0010 | 1.2789 | 1.1261 | 2.0440 | 0.0012 | 1.1926 |
10,000 | ||||||||
---|---|---|---|---|---|---|---|---|
FVM (13) | FVM [21] | FVM (13) | FVM [21] | |||||
Max-Error | Order | Max-Error | Order | Max-Error | Order | Max-Error | Order | |
1/10 | 0.0026 | – | 0.0124 | – | 0.0026 | – | 0.0124 | – |
1/20 | 7.2459 | 1.8364 | 0.0063 | 0.9727 | 7.2459 | 1.83 | 0.0063 | 0.9712 |
1/40 | 1.9495 | 1.8941 | 0.0032 | 0.9844 | 1.9495 | 1.8967 | 0.0032 | 0.9832 |
1/80 | 5.4858 | 1.8293 | 0.0016 | 0.9963 | 5.1496 | 1.9425 | 0.0016 | 0.99151 |
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Hussain, A.; Zheng, Z.; Anley, E.F. Numerical Analysis of Convection–Diffusion Using a Modified Upwind Approach in the Finite Volume Method. Mathematics 2020, 8, 1869. https://doi.org/10.3390/math8111869
Hussain A, Zheng Z, Anley EF. Numerical Analysis of Convection–Diffusion Using a Modified Upwind Approach in the Finite Volume Method. Mathematics. 2020; 8(11):1869. https://doi.org/10.3390/math8111869
Chicago/Turabian StyleHussain, Arafat, Zhoushun Zheng, and Eyaya Fekadie Anley. 2020. "Numerical Analysis of Convection–Diffusion Using a Modified Upwind Approach in the Finite Volume Method" Mathematics 8, no. 11: 1869. https://doi.org/10.3390/math8111869
APA StyleHussain, A., Zheng, Z., & Anley, E. F. (2020). Numerical Analysis of Convection–Diffusion Using a Modified Upwind Approach in the Finite Volume Method. Mathematics, 8(11), 1869. https://doi.org/10.3390/math8111869