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

Multi-scale analysis for highly anisotropic parabolic problems

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.

Citation: Thomas Blanc, Mihaï Bostan. Multi-scale analysis for highly anisotropic parabolic problems. Discrete and 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.

[2]

D. S. BalsanaD. 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. 

[3]

T. BlancM. 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.

[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.

[5]

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

[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.

[7]

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

[8]

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

[9]

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

[10]

P. DegondF. 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.

[11]

R. EymardT. 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.

[12]

F. FilbetC. 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.

[13]

E. FreireA. 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.

[14]

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

[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.

[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.

[17]

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

[18]

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

[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.

[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.

[21]

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

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.

[2]

D. S. BalsanaD. 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. 

[3]

T. BlancM. 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.

[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.

[5]

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

[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.

[7]

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

[8]

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

[9]

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

[10]

P. DegondF. 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.

[11]

R. EymardT. 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.

[12]

F. FilbetC. 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.

[13]

E. FreireA. 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.

[14]

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

[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.

[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.

[17]

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

[18]

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

[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.

[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.

[21]

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

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