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January  2020, 25(1): 335-399. doi: 10.3934/dcdsb.2019186

Multi-scale analysis for highly anisotropic parabolic problems

Thomas Blanc and  Mihaï Bostan ,

Aix Marseille Université, CNRS, Centrale Marseille, Institut de Mathématiques de Marseille UMR 7373, Château Gombert 39 rue F. Joliot Curie, Marseille, 13453, FRANCE

* Corresponding author: Mihaï Bostan

Received  February 2018 Revised  February 2019 Published  January 2020 Early access  September 2019

We focus on the asymptotic behavior of strongly anisotropic parabolic problems. We concentrate on heat equations, whose diffusion matrix fields have disparate eigenvalues. We establish strong convergence results toward a profile. Under suitable smoothness hypotheses, by introducing an appropriate corrector term, we estimate the convergence rate. The arguments rely on two-scale analysis, based on average operators with respect to unitary groups.

Keywords: Parabolic problems, average operators, ergodic means, unitary groups, homogenization.
Mathematics Subject Classification: Primary: 35K05; Secondary: 37A10.
Citation: Thomas Blanc, Mihaï Bostan. Multi-scale analysis for highly anisotropic parabolic problems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 335-399. doi: 10.3934/dcdsb.2019186
References:
[1]

L. Agelas and R. Masson, Convergence of the finite volume MPFA O scheme for heterogeneous anisotropic diffusion problems on general meshes, C. R. Math. Acad. Sci. Paris, 346 (2008), 1007-1012.  doi: 10.1016/j.crma.2008.07.015.  Google Scholar

[2]

D. S. Balsana, D. A. Tilley and C. J. Howk, Simulating anisotropic thermal conduction in supernova remnants-Ⅰ., Numerical methods, Monthly Notices of Royal Astronomical Society, 386 (2008), 627-641.   Google Scholar

[3]

T. Blanc, M. Bostan and F. Boyer, Asymptotic analysis of parabolic equations with stiff transport terms by a multi-scale approach, Discrete Contin. Dyn. Syst. Ser. A, 37 (2017), 4637-4676.  doi: 10.3934/dcds.2017200.  Google Scholar

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M. Bostan, Transport equations with disparate advection fields. Application to the gyrokinetic models in plasma physics, J. Differential Equations, 249 (2010), 1620-1663.  doi: 10.1016/j.jde.2010.07.010.  Google Scholar

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M. Bostan, Strongly anisotropic diffusion problems; asymptotic analysis, J. Differential Equations, 256 (2016), 1043-1092.  doi: 10.1016/j.jde.2013.10.008.  Google Scholar

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M. Bostan, Multi-scale analysis for linear first order PDEs. The finite Larmor radius regime, SIAM J. Math. Anal., 48 (2016), 2133-2188.  doi: 10.1137/15M1033034.  Google Scholar

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S. I. Braginskii, Transport Processes in a Plasma, M.A. Leontovich, Reviews of Plasma Physics, Consultants Bureau, New York, 1965. Google Scholar

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C. Corduneanu, Almost Periodic Oscillations and Waves, Springer, 2009. doi: 10.1007/978-0-387-09819-7.  Google Scholar

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R. Dautray and J.-L. Lions, Analyse Mathématique et Calcul Numérique Pour Les Sciences et Les Techniques, vol. 8, Masson, 1988.  Google Scholar

[10]

P. Degond, F. Deluzet and C. Negulescu, An asymptotic preserving scheme for strongly anisotropic elliptic problems, Multiscale Model. Simul., 8 (2009/10), 645-666.  doi: 10.1137/090754200.  Google Scholar

[11]

R. Eymard, T. Gallouët and R. Herbin, Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces, IMA J. Numer. Anal., 30 (2010), 1009-1043.  doi: 10.1093/imanum/drn084.  Google Scholar

[12]

F. Filbet, C. Negulescu and C. Yang, Numerical study of a nonlinear heat equation for plasma physics, Int. J. Comput. Math., 89 (2012), 1060-1082.  doi: 10.1080/00207160.2012.679732.  Google Scholar

[13]

E. Freire, A. Gasull and A. Guillamon, A characterization of isochronous centers in terms of symmetries, Rev. Mat. Iberoamericana, 20 (2004), 205-222.  doi: 10.4171/RMI/386.  Google Scholar

[14]

J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, vol. Ⅰ, Springer Berlin Heidelberg, 1972.  Google Scholar

[15]

P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell., 12 (1990), 629-639.  doi: 10.1109/34.56205.  Google Scholar

[16]

J. Quah and D. Margetis, Anisotropic diffusion in continuum relaxation of stepped crystal surfaces, J. Phys. A, 41 (2008), 235004, 18pp. doi: 10.1088/1751-8113/41/23/235004.  Google Scholar

[17]

M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. Ⅰ, Functional Analysis, Academic Press, 1980.  Google Scholar

[18]

M. Sabatini, Characterizing isochronous centers by Lie brackets, Differential Equations Dyn. Syst., 5 (1997), 91-99.   Google Scholar

[19]

P. Sharma and G. W. Hammett, A fast semi-implicit method for anisotropic diffusion, J. Comput. Phys., 230 (2011), 4899-4909.  doi: 10.1016/j.jcp.2011.03.009.  Google Scholar

[20]

P. Sharma and G. W. Hammett, Preserving monotonicity in anisotropic diffusion, J. Comput. Phys., 227 (2007), 123-142.  doi: 10.1016/j.jcp.2007.07.026.  Google Scholar

[21]

J. Weickert, Anisotropic Diffusion in Image Processing, Teubner, Stuttgart 1998.  Google Scholar

show all references

References:
[1]

L. Agelas and R. Masson, Convergence of the finite volume MPFA O scheme for heterogeneous anisotropic diffusion problems on general meshes, C. R. Math. Acad. Sci. Paris, 346 (2008), 1007-1012.  doi: 10.1016/j.crma.2008.07.015.  Google Scholar

[2]

D. S. Balsana, D. A. Tilley and C. J. Howk, Simulating anisotropic thermal conduction in supernova remnants-Ⅰ., Numerical methods, Monthly Notices of Royal Astronomical Society, 386 (2008), 627-641.   Google Scholar

[3]

T. Blanc, M. Bostan and F. Boyer, Asymptotic analysis of parabolic equations with stiff transport terms by a multi-scale approach, Discrete Contin. Dyn. Syst. Ser. A, 37 (2017), 4637-4676.  doi: 10.3934/dcds.2017200.  Google Scholar

[4]

M. Bostan, Transport equations with disparate advection fields. Application to the gyrokinetic models in plasma physics, J. Differential Equations, 249 (2010), 1620-1663.  doi: 10.1016/j.jde.2010.07.010.  Google Scholar

[5]

M. Bostan, Strongly anisotropic diffusion problems; asymptotic analysis, J. Differential Equations, 256 (2016), 1043-1092.  doi: 10.1016/j.jde.2013.10.008.  Google Scholar

[6]

M. Bostan, Multi-scale analysis for linear first order PDEs. The finite Larmor radius regime, SIAM J. Math. Anal., 48 (2016), 2133-2188.  doi: 10.1137/15M1033034.  Google Scholar

[7]

S. I. Braginskii, Transport Processes in a Plasma, M.A. Leontovich, Reviews of Plasma Physics, Consultants Bureau, New York, 1965. Google Scholar

[8]

C. Corduneanu, Almost Periodic Oscillations and Waves, Springer, 2009. doi: 10.1007/978-0-387-09819-7.  Google Scholar

[9]

R. Dautray and J.-L. Lions, Analyse Mathématique et Calcul Numérique Pour Les Sciences et Les Techniques, vol. 8, Masson, 1988.  Google Scholar

[10]

P. Degond, F. Deluzet and C. Negulescu, An asymptotic preserving scheme for strongly anisotropic elliptic problems, Multiscale Model. Simul., 8 (2009/10), 645-666.  doi: 10.1137/090754200.  Google Scholar

[11]

R. Eymard, T. Gallouët and R. Herbin, Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces, IMA J. Numer. Anal., 30 (2010), 1009-1043.  doi: 10.1093/imanum/drn084.  Google Scholar

[12]

F. Filbet, C. Negulescu and C. Yang, Numerical study of a nonlinear heat equation for plasma physics, Int. J. Comput. Math., 89 (2012), 1060-1082.  doi: 10.1080/00207160.2012.679732.  Google Scholar

[13]

E. Freire, A. Gasull and A. Guillamon, A characterization of isochronous centers in terms of symmetries, Rev. Mat. Iberoamericana, 20 (2004), 205-222.  doi: 10.4171/RMI/386.  Google Scholar

[14]

J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, vol. Ⅰ, Springer Berlin Heidelberg, 1972.  Google Scholar

[15]

P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell., 12 (1990), 629-639.  doi: 10.1109/34.56205.  Google Scholar

[16]

J. Quah and D. Margetis, Anisotropic diffusion in continuum relaxation of stepped crystal surfaces, J. Phys. A, 41 (2008), 235004, 18pp. doi: 10.1088/1751-8113/41/23/235004.  Google Scholar

[17]

M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. Ⅰ, Functional Analysis, Academic Press, 1980.  Google Scholar

[18]

M. Sabatini, Characterizing isochronous centers by Lie brackets, Differential Equations Dyn. Syst., 5 (1997), 91-99.   Google Scholar

[19]

P. Sharma and G. W. Hammett, A fast semi-implicit method for anisotropic diffusion, J. Comput. Phys., 230 (2011), 4899-4909.  doi: 10.1016/j.jcp.2011.03.009.  Google Scholar

[20]

P. Sharma and G. W. Hammett, Preserving monotonicity in anisotropic diffusion, J. Comput. Phys., 227 (2007), 123-142.  doi: 10.1016/j.jcp.2007.07.026.  Google Scholar

[21]

J. Weickert, Anisotropic Diffusion in Image Processing, Teubner, Stuttgart 1998.  Google Scholar

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