Advection-diffusion equations are frequently encountered in many fields and have become a very active research area. However, in most cases, these equations concern different time and space scales, which make it impossible to derive explicit solutions to these equations. To overcome this difficulty, the theory of homogenization aims to approximate the original differential equation with rapidly oscillating coefficients by an effective homogenized equation with constant or slowly varying coefficients. The homogenized equation is often quite suitable for theoretical analysis or numerical methods. This paper investigates the homogenization principle of an advection-diffusion partial differential equation. The novelty of the parabolic partial differential equation we consider is that the advection term in the equation is two-scaled, which is rarely considered by others for the homogenization of advection-diffusion equation. Under certain proper assumptions on the coefficient functions of the original advection-diffusion partial differential equation, which ensure the variable elimination, we derive the homogenized equation, which is also an advection-diffusion equation, by the technique of multi-scale expansion. It is shown that the coefficient functions of the original two-scaled equation have different influence on the coefficient functions of the homogenized equation.
Published in | Mathematics Letters (Volume 8, Issue 3) |
DOI | 10.11648/j.ml.20220803.11 |
Page(s) | 43-47 |
Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
Copyright |
Copyright © The Author(s), 2023. Published by Science Publishing Group |
Homogenization, Partial Differential Equation, Multi-scale Expansion
[1] | G. Allaire, Shape optimization by the homogenization method, volume 146 of Applied Mathematical Sciences. Springer-Verlag, New York, 2002. |
[2] | I. Babuka, Homogenization and its application. Mathematical and computational problems. Numerical solution of partial differential equations-III. Academic Press, 1976. 89-116. |
[3] | I. Babuka, Homogenization approach in engineering. Computing methods in applied sciences and engineering. Springer, Berlin, Heidelberg, 1976. 137-153. |
[4] | I. Babuka, Solution of interface problems by homogenization. I. SIAM J. Math. Anal. 7 (5) (1975), 603-634. |
[5] | I. Babuka, Solution of interface problems by homogenization. II. SIAM J. Math. Anal. 7 (5) (1975), 635-645. |
[6] | N. Bakhvalov and P. Grigory, Homogenisation: averaging processes in periodic media: mathematical problems in the mechanics of composite materials. Vol. 36. Springer Science & Business Media, 2012. |
[7] | A. Bensoussan, J. L. Lions, and P. George, Asymptotic analysis for periodic structures. Vol. 374. American Mathematical Soc., 2011. |
[8] | N. N. Bogoliubov and Y. A. Mitropolsky, Asymptotic Methods in the Theory of Non-linear Oscillations, Gordon and Breach Science Publishers, New York, 1961. |
[9] | D. Cioranescu and P. Donato, An introduction to homogenization. Vol. 17. Oxford: Oxford university press, 1999. |
[10] | D. Cioranescu and J. S. J. Pauline, Homogenization of reticulated structures, volume 136 of Applied Mathematical Sciences. Springer-Verlag, New York, 1999. |
[11] | G. Falkovich, K.Gawedzki, and M.Vergassola, Particles and fields in fluid turbulance. Rev. Modern Phys. 73 (4) (2001), 913-975. |
[12] | A. Fannjiang and G. C. Papanicolaou, Convection enhanced diffusion for periodic flows. SIAM J. APPL. MATH. 54 (1994), 333-408. |
[13] | A. Fannjiang and G. C. Papanicolaou, Convection enhanced diffusion for random flows. J. Stat. Phys. 88 (1997), 1033-1076. |
[14] | J. Garnier, Homogenization in a periodic and time- dependent potential. SIAM J.APPL.MATH. 57 (1) (1997), 95-111. |
[15] | V. V. Jikov, S. M .Kozlov, and O. A. Oleinik, Homogenization of differential operators and integral functionals. Springer, Berlin, 1994. |
[16] | J. B. Keller, Darcy’s law for flow in porous media and the two-space method. Nonlinear partial differential equations in engineering and applied science. Routledge, 2017. 429-443. |
[17] | J. B. Keller, Effective behavior of heterogeneous media. Statistical mechanics and statistical methods in theory and application. Springer, Boston, MA, 1977. 631-644. |
[18] | A. J. Majda and P. R. Kramer, Simplified models for turbulent diffusion: Theory, numerical modelling and physical phenomena. Physics Reports. 314 (1999), 237- 574. |
[19] | D. W. McLaughlin, G. C. Papanicolaou, and O. R. Pironneau, Convection of microstructure and related problems. SIAM J. APPL. MATH. 45 (1985), 780-797. |
[20] | R. M. McLaughlin and R. M. Forest, An anelastic, scale-separated model for mixing, with application to atmospheric transport phenomena. Phys. Fluids. 11 (4) (1999), 880-892. |
[21] | I. Mezic, J. F. Brady, and S. Wiggins, Maximal effective diffusitity for time-periodic incompressible fluid flows. SIAM J. APPL. MATH. 56 (1) (1996), 40-56. |
[22] | O. A. Olenik, A. S. Shamaev, and G. A. Yosifian, Mathematical problems in elasticity and homogenization, volume 26 of Studies in Mathematics and ite Applications. North-Holland Publishing Co., Amsterdam, 1992. |
[23] | S. Olla, Homogenization of diffusion processes in random fields. Lecture Notes, 1994. |
[24] | G. A. Pavliotis, A multiscale approach to Brownian motors. Phys. Lett. A. 344 (2005), 331-345. |
[25] | G. A. Pavliotis and A. M. Stuart, Multiscale Methods: Averaging and Homogenization. Springer, New York, 2008. |
[26] | E. Sánchez-Palencia, Nonhomogeneous media and vibration theory, volume 127 of Lecture Notes in Physics. Springer-Verlag, Berlin, 1980. |
[27] | M. Vergassola and M. Avellaneda, Scalar transport in compressible flow. Phys. D. 106 (1-2) (1997), 148-166. |
APA Style
Tao Jiang. (2023). Homogenization for a Parabolic Partial Differential Equation with Two-scaled Advection. Mathematics Letters, 8(3), 43-47. https://doi.org/10.11648/j.ml.20220803.11
ACS Style
Tao Jiang. Homogenization for a Parabolic Partial Differential Equation with Two-scaled Advection. Math. Lett. 2023, 8(3), 43-47. doi: 10.11648/j.ml.20220803.11
AMA Style
Tao Jiang. Homogenization for a Parabolic Partial Differential Equation with Two-scaled Advection. Math Lett. 2023;8(3):43-47. doi: 10.11648/j.ml.20220803.11
@article{10.11648/j.ml.20220803.11, author = {Tao Jiang}, title = {Homogenization for a Parabolic Partial Differential Equation with Two-scaled Advection}, journal = {Mathematics Letters}, volume = {8}, number = {3}, pages = {43-47}, doi = {10.11648/j.ml.20220803.11}, url = {https://doi.org/10.11648/j.ml.20220803.11}, eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20220803.11}, abstract = {Advection-diffusion equations are frequently encountered in many fields and have become a very active research area. However, in most cases, these equations concern different time and space scales, which make it impossible to derive explicit solutions to these equations. To overcome this difficulty, the theory of homogenization aims to approximate the original differential equation with rapidly oscillating coefficients by an effective homogenized equation with constant or slowly varying coefficients. The homogenized equation is often quite suitable for theoretical analysis or numerical methods. This paper investigates the homogenization principle of an advection-diffusion partial differential equation. The novelty of the parabolic partial differential equation we consider is that the advection term in the equation is two-scaled, which is rarely considered by others for the homogenization of advection-diffusion equation. Under certain proper assumptions on the coefficient functions of the original advection-diffusion partial differential equation, which ensure the variable elimination, we derive the homogenized equation, which is also an advection-diffusion equation, by the technique of multi-scale expansion. It is shown that the coefficient functions of the original two-scaled equation have different influence on the coefficient functions of the homogenized equation.}, year = {2023} }
TY - JOUR T1 - Homogenization for a Parabolic Partial Differential Equation with Two-scaled Advection AU - Tao Jiang Y1 - 2023/02/01 PY - 2023 N1 - https://doi.org/10.11648/j.ml.20220803.11 DO - 10.11648/j.ml.20220803.11 T2 - Mathematics Letters JF - Mathematics Letters JO - Mathematics Letters SP - 43 EP - 47 PB - Science Publishing Group SN - 2575-5056 UR - https://doi.org/10.11648/j.ml.20220803.11 AB - Advection-diffusion equations are frequently encountered in many fields and have become a very active research area. However, in most cases, these equations concern different time and space scales, which make it impossible to derive explicit solutions to these equations. To overcome this difficulty, the theory of homogenization aims to approximate the original differential equation with rapidly oscillating coefficients by an effective homogenized equation with constant or slowly varying coefficients. The homogenized equation is often quite suitable for theoretical analysis or numerical methods. This paper investigates the homogenization principle of an advection-diffusion partial differential equation. The novelty of the parabolic partial differential equation we consider is that the advection term in the equation is two-scaled, which is rarely considered by others for the homogenization of advection-diffusion equation. Under certain proper assumptions on the coefficient functions of the original advection-diffusion partial differential equation, which ensure the variable elimination, we derive the homogenized equation, which is also an advection-diffusion equation, by the technique of multi-scale expansion. It is shown that the coefficient functions of the original two-scaled equation have different influence on the coefficient functions of the homogenized equation. VL - 8 IS - 3 ER -