Abstract
Bu çalışmada, konveksiyon-difüzyon denklemi; birinci dereceden yukarı yelpaze, ikinci dereceden merkezi fark alma, kübik (kısmen yukarı yelpaze uygulanmış) ve kübik-TVD (Toplam Varyasyon Azaltma) teknikleri olmak üzere dört farklı uzay ayrıklaştırma yöntemi kullanılarak sayısal olarak çözülmüştür. Tüm yöntemler analitik çözümle karşılaştırılmıştır. Birinci dereceden yöntem, sayısal dağılım nedeniyle analitik çözüme yakın değildir. Daha yüksek mertebeden teknikler sayısal dağılımı azaltmaktadır, ancak bir diğer sayısal hataya, fiziksel olmayan salınımlara sebep olmaktadırlar. Bu çalışma, istenmeyen salınımları ortadan kaldırmak için kübik-TVD yöntemine yenilikçi bir yaklaşım önermektedir. Önerilen model, daha önce geliştirilen tekniklere kıyasla sayısal hataları önemli ölçüde azaltır. Ayrıca sunulan modelin sayısal sonuçları analitik çözüme oldukça yakındır. Son olarak, diğer araştırmacıların işini kolaylaştırmak için konveksiyon-difüzyon denklemi için sayısal ve analitik çözümlerin tüm Matlab kodları Ek'e dahil edilmiştir.
References
- Ertekin, T., Abou-Kassem, J. H., & King, G. R. (2001). Basic applied reservoir simulation (Vol. 7). Richardson, TX: Society of Petroleum Engineers.
- Harten, A. (1984). On a class of high resolution total-variation-stable finite-difference schemes. SIAM Journal on Numerical Analysis, 21(1), 1-23.
- Kamalyar, K., Kharrat, R., & Nikbakht, M. (2014). Numerical Aspects of the Convection–Dispersion Equation. Petroleum science and technology, 32(14), 1729-1762.
- Kurganov, A., & Tadmor, E. (2000). New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations. Journal of Computational Physics, 160(1), 241-282.
- Leonard, B. P. (1979). A stable and accurate convective modelling procedure based on quadratic upstream interpolation. Computer methods in applied mechanics and engineering, 19(1), 59-98.
- Mazumder, S. (2015). Numerical methods for partial differential equations: finite difference and finite volume methods. Academic Press.
- Peaceman, D. W. (2000). Fundamentals of numerical reservoir simulation. Elsevier.
- Peng, Y., Liu, C., & Shi, L. (2013, August). Soution of Convection-Diffusion Equations. In International Conference on Information Computing and Applications (pp. 546-555). Springer, Berlin, Heidelberg.
- Sarra, S. A. (2003). The method of characteristics with applications to conservation laws. Journal of Online mathematics and its Applications, 3, 1-16.
- Sweby, P. K. (1984). High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM journal on numerical analysis, 21(5), 995-1011.
- Wolcott, D. S., Kazemi, H., & Dean, R. H. (1996, October). A practical method for minimizing the grid orientation effect in reservoir simulation. In SPE annual technical conference and exhibition. OnePetro.
Abstract
In this study, convection-diffusion equation is solved numerically using four different space discretization methods namely first-order upwinding, second-order central difference, cubic (partially upwinded) and cubic-TVD (Total Variation Diminishing) techniques. All methods are compared with the analytical solution. The first-order method is not close to the analytical solution due to the numerical dispersion. The higher-order techniques reduce numerical dispersion. However, they cause another numerical error, unphysical oscillation. This study proposes an innovative approach on cubic-TVD method to eliminate undesired oscillations. Proposed model decreases numerical errors significantly compared to previously developed techniques. Moreover, numerical results of presented model quite close to the analytical solution. Finally, all Matlab codes of numerical and analytical solutions for convection-diffusion equation are added to Appendix in order to facilitate other researchers’ work.
References
- Ertekin, T., Abou-Kassem, J. H., & King, G. R. (2001). Basic applied reservoir simulation (Vol. 7). Richardson, TX: Society of Petroleum Engineers.
- Harten, A. (1984). On a class of high resolution total-variation-stable finite-difference schemes. SIAM Journal on Numerical Analysis, 21(1), 1-23.
- Kamalyar, K., Kharrat, R., & Nikbakht, M. (2014). Numerical Aspects of the Convection–Dispersion Equation. Petroleum science and technology, 32(14), 1729-1762.
- Kurganov, A., & Tadmor, E. (2000). New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations. Journal of Computational Physics, 160(1), 241-282.
- Leonard, B. P. (1979). A stable and accurate convective modelling procedure based on quadratic upstream interpolation. Computer methods in applied mechanics and engineering, 19(1), 59-98.
- Mazumder, S. (2015). Numerical methods for partial differential equations: finite difference and finite volume methods. Academic Press.
- Peaceman, D. W. (2000). Fundamentals of numerical reservoir simulation. Elsevier.
- Peng, Y., Liu, C., & Shi, L. (2013, August). Soution of Convection-Diffusion Equations. In International Conference on Information Computing and Applications (pp. 546-555). Springer, Berlin, Heidelberg.
- Sarra, S. A. (2003). The method of characteristics with applications to conservation laws. Journal of Online mathematics and its Applications, 3, 1-16.
- Sweby, P. K. (1984). High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM journal on numerical analysis, 21(5), 995-1011.
- Wolcott, D. S., Kazemi, H., & Dean, R. H. (1996, October). A practical method for minimizing the grid orientation effect in reservoir simulation. In SPE annual technical conference and exhibition. OnePetro.
Details
| Primary Language | English |
|---|---|
| Subjects | Engineering |
| Journal Section | Articles |
| Authors | |
| Publication Date | December 15, 2022 |
| Published in Issue | Year 2022 Volume: 12 Issue: 2 |
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