Araştırma Makalesi
Yıl 2023,
Cilt: 72 Sayı: 4, 1077 - 1093, 29.12.2023
Öz
In this study, we first consider the time-relaxation model, which consists of adding the term $\kappa \left( u-\overline{u}\right) $ to the heat equation. Then, an explicit discretization scheme for the model is introduced to find the finite difference solutions. We first obtain the solutions by using the scheme and then investigate the method’s consistency, stability, and convergence properties. We prove that the method is consistent and unconditionally stable for any given value of $r$ and appropriate values of $\kappa$ and $\delta$. As a result, the method obtained by adding the time relaxation term to the first-order finite-difference explicit method behaves like the second-order implicit method. Finally, we apply the method to some test examples.
Anahtar Kelimeler
Heat equation time-relaxation model differential filter finite difference method
Kaynakça
- Adams, N. A., Stolz, S., Deconvolution methods for subgrid-scale approximation in LES, in Modern Simulation Strategies for Turbulent Flow (eds. B. Geurts and R.T. Edwards), (2001), 21–41.
- Asselin, R., Frequency filter for time integrations, Mon. Weather Rev., 100 (1972), 487–490. https://doi.org/10.1175/1520-0493(1972)100<0487:FFFTI>2.3.CO;2
- Burden, R. L., Faires, J. D., Numerical Analysis, Brooks/Cole, Pacific Grove, 9th Edition, 2011.
- Cibik, A., Eroglu, F. G., Kaya, S., Numerical analysis of an efficient second order time filtered backward Euler method for MHD equations, Journal of Scientific Computing, 82(2) (2020), 1–25. https://doi.org/10.48550/arXiv.1906.07922
- Cooper, J., Introduction to Partial Differential Equations with Matlab, Birkhauser, Bosten, 1998.
- Ervin, V. J., Layton, W., Neda, M., Numerical analysis of a higher order time relaxation model of fluids, Int. J. Numer. Anal. Model., 4 (2007), 648–670.
- Fletcher, C. A. J., Computational Techniques for Fluid Dynamics, Springer-Verlag, Berlin, 1998.
- Gresho, P. M., Sani, R. L., Incompressible Flow and the Finite Element Method, John Wiley & Sons, Inc, 1998.
- Guzel, A., Layton, W., Time filters increase accuracy of the fully implicit method, BIT Numer. Math., 58(3) (2018), 301–315. DOI:10.1007/s10543-018-0695-z
- Isik, O. R., Spin up problem and accelerating convergence to steady state, Applied Mathematical Modelling, 37(5) (2013), 3242–3253. https://doi.org/10.1016/j.apm.2012.07.033
- Isik, O. R., Takhirov, A., Zheng, H., Second order time relaxation model for accelerating convergence to steady-state equilibrium for Navier-Stokes equations, Applied Numerical Mathematics, 119 (2017), 67–78. https://doi.org/10.1016/j.apnum.2017.03.016
- Isik, O. R., Yuksel, G., Demir, B., Analysis of second order and unconditonally stable for BDF2-AB2 method for the Navier-Stokes Equations with nonlinear time relaxation, Numerical Methods for Partial Differential Equations, 34(6) (2018), 2060–2078. https://doi.org/10.1002/num.22276
- Layton, W., Neda, M., Truncation of scales by time relaxation, J. Math. Anal. Appl., 325 (2007), 788–807. https://doi.org/10.1016/j.jmaa.2006.02.014
- Layton, W., Pruett, C. D., Rebholz, L. G. Temporally regularized direct numerical simulation, Appl. Math. Comput., 216 (2010), 3728–3738. DOI:10.1016/j.amc.2010.05.031
- Li, Y., Trenchea, C., A higher-order Robert-Asselin type time filter, J. Comput. Phys., 259 (2014), 23–32. https://doi.org/10.1016/j.jcp.2013.11.022
- Morton, K. W., Mayers, D. F., Numerical Solution of Partial Differential Equations: An Introduction, Cambridge University Press, 1994.
- Neda, M., Discontinuous time relaxation method for the time-dependent Navier-Stokes equations, Advance in Numerical Analysis, 2010, Article ID 419021, 21 pages, (2010). https://doi.org/10.1155/2010/419021
- Pruett, C. D., Gatski, T. B., Grosch, C. E., Thacker, W. D., The temporally filtered Navier–Stokes equations: properties of the residual-stres, Phys. Fluids, 15 (2003), 2127–2140. https://doi.org/10.1063/1.1582858
- Rosenau, P., Extending hydrodynamics via the regularization of the Chapman-Enskog expansion, Phys. Rev., A40(7193) (1989). https://doi.org/10.1103/PhysRevA.40.7193
- Schochet, S., Tadmor, E., The regularized Chapman-Enskog expansion for scalar conservation laws, Arch. Ration. Mech. Anal., 119(95) (1992). https://doi.org/10.1007/BF00375117
- Smith, G. D., Numerical Solution of Partial Differential Equations: Finite Difference Methods, Oxford University Press, 1985.
- Stolz, S., Adams, N. A., Kleiser, L., The approximate deconvolution model for LES of compressible flows and its application to shock-turbulent boundary layer interaction, Phys. Fluids, 13(2985) (2001). https://doi.org/10.1063/1.1397277
- Williams, P. D., A proposed modification to the Robert-Asselin time filter, Mon. Weather Re., 137(8) (2009), 2538–2546. https://doi.org/10.1175/2009MWR2724.1
- Williams, P. D., The RAW filter: An improvement to the Robert-Asselin filter in semiimplicit integrations, Mon. Weather Rev., 139(6) (2011), 1996–2007. https://doi.org/10.1175/2010MWR3601.1
Yıl 2023,
Cilt: 72 Sayı: 4, 1077 - 1093, 29.12.2023
Öz
Kaynakça
- Adams, N. A., Stolz, S., Deconvolution methods for subgrid-scale approximation in LES, in Modern Simulation Strategies for Turbulent Flow (eds. B. Geurts and R.T. Edwards), (2001), 21–41.
- Asselin, R., Frequency filter for time integrations, Mon. Weather Rev., 100 (1972), 487–490. https://doi.org/10.1175/1520-0493(1972)100<0487:FFFTI>2.3.CO;2
- Burden, R. L., Faires, J. D., Numerical Analysis, Brooks/Cole, Pacific Grove, 9th Edition, 2011.
- Cibik, A., Eroglu, F. G., Kaya, S., Numerical analysis of an efficient second order time filtered backward Euler method for MHD equations, Journal of Scientific Computing, 82(2) (2020), 1–25. https://doi.org/10.48550/arXiv.1906.07922
- Cooper, J., Introduction to Partial Differential Equations with Matlab, Birkhauser, Bosten, 1998.
- Ervin, V. J., Layton, W., Neda, M., Numerical analysis of a higher order time relaxation model of fluids, Int. J. Numer. Anal. Model., 4 (2007), 648–670.
- Fletcher, C. A. J., Computational Techniques for Fluid Dynamics, Springer-Verlag, Berlin, 1998.
- Gresho, P. M., Sani, R. L., Incompressible Flow and the Finite Element Method, John Wiley & Sons, Inc, 1998.
- Guzel, A., Layton, W., Time filters increase accuracy of the fully implicit method, BIT Numer. Math., 58(3) (2018), 301–315. DOI:10.1007/s10543-018-0695-z
- Isik, O. R., Spin up problem and accelerating convergence to steady state, Applied Mathematical Modelling, 37(5) (2013), 3242–3253. https://doi.org/10.1016/j.apm.2012.07.033
- Isik, O. R., Takhirov, A., Zheng, H., Second order time relaxation model for accelerating convergence to steady-state equilibrium for Navier-Stokes equations, Applied Numerical Mathematics, 119 (2017), 67–78. https://doi.org/10.1016/j.apnum.2017.03.016
- Isik, O. R., Yuksel, G., Demir, B., Analysis of second order and unconditonally stable for BDF2-AB2 method for the Navier-Stokes Equations with nonlinear time relaxation, Numerical Methods for Partial Differential Equations, 34(6) (2018), 2060–2078. https://doi.org/10.1002/num.22276
- Layton, W., Neda, M., Truncation of scales by time relaxation, J. Math. Anal. Appl., 325 (2007), 788–807. https://doi.org/10.1016/j.jmaa.2006.02.014
- Layton, W., Pruett, C. D., Rebholz, L. G. Temporally regularized direct numerical simulation, Appl. Math. Comput., 216 (2010), 3728–3738. DOI:10.1016/j.amc.2010.05.031
- Li, Y., Trenchea, C., A higher-order Robert-Asselin type time filter, J. Comput. Phys., 259 (2014), 23–32. https://doi.org/10.1016/j.jcp.2013.11.022
- Morton, K. W., Mayers, D. F., Numerical Solution of Partial Differential Equations: An Introduction, Cambridge University Press, 1994.
- Neda, M., Discontinuous time relaxation method for the time-dependent Navier-Stokes equations, Advance in Numerical Analysis, 2010, Article ID 419021, 21 pages, (2010). https://doi.org/10.1155/2010/419021
- Pruett, C. D., Gatski, T. B., Grosch, C. E., Thacker, W. D., The temporally filtered Navier–Stokes equations: properties of the residual-stres, Phys. Fluids, 15 (2003), 2127–2140. https://doi.org/10.1063/1.1582858
- Rosenau, P., Extending hydrodynamics via the regularization of the Chapman-Enskog expansion, Phys. Rev., A40(7193) (1989). https://doi.org/10.1103/PhysRevA.40.7193
- Schochet, S., Tadmor, E., The regularized Chapman-Enskog expansion for scalar conservation laws, Arch. Ration. Mech. Anal., 119(95) (1992). https://doi.org/10.1007/BF00375117
- Smith, G. D., Numerical Solution of Partial Differential Equations: Finite Difference Methods, Oxford University Press, 1985.
- Stolz, S., Adams, N. A., Kleiser, L., The approximate deconvolution model for LES of compressible flows and its application to shock-turbulent boundary layer interaction, Phys. Fluids, 13(2985) (2001). https://doi.org/10.1063/1.1397277
- Williams, P. D., A proposed modification to the Robert-Asselin time filter, Mon. Weather Re., 137(8) (2009), 2538–2546. https://doi.org/10.1175/2009MWR2724.1
- Williams, P. D., The RAW filter: An improvement to the Robert-Asselin filter in semiimplicit integrations, Mon. Weather Rev., 139(6) (2011), 1996–2007. https://doi.org/10.1175/2010MWR3601.1
Toplam 24 adet kaynakça vardır.
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Konular | Uygulamalı Matematik |
| Bölüm | Research Article |
| Yazarlar | |
| Yayımlanma Tarihi | 29 Aralık 2023 |
| Gönderilme Tarihi | 29 Ocak 2023 |
| Kabul Tarihi | 18 Temmuz 2023 |
| Yayımlandığı Sayı | Yıl 2023 Cilt: 72 Sayı: 4 |
Kaynak Göster
Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.
