American Institute of Mathematical Sciences

October  2016, 9(5): i-ix. doi: 10.3934/dcdss.201605i

Homogenization-based numerical methods

 1 Université de Bretagne-Sud, UMR 6205, LMBA, F-56000 Vannes

Published  October 2016

This note recalls what are "Homogenization-Based Numerical Methods". Then it introduces the papers of this Special Issue. In a third section it advocates for building a project in order to build "Homogenization- Based Software for Simulation of Multi-Scale Complex Systems".
Citation: Emmanuel Frénod. Homogenization-based numerical methods. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : i-ix. doi: 10.3934/dcdss.201605i
References:
 [1] A. Abdulle, Y. Bai and G. Vilmart, Reduced basis finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, 8 (2015), 91-118. doi: 10.3934/dcdss.2015.8.91. [2] P. Ailliot, E. Frénod and V. Monbet, Long term object drift forecast in the ocean with tide and wind, Multiscale Modeling and Simulations, 5 (2006), 514-531. doi: 10.1137/050639727. [3] G. Allaire and R. Brizzi, A multiscale finite element method for numerical homogenization, SIAM Multiscale Modeling and Simulations, 4 (2005), 790-812. doi: 10.1137/040611239. [4] A. Back and E. Frénod, Geometric two-scale convergence on manifold and applications to the Vlasov equation, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, 8 (2015), 223-241. doi: 10.3934/dcdss.2015.8.223. [5] J.-P. Bernard, E. Frénod and A. Rousseau, Paralic confinement computations in coastal environment with interlocked areas, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, 8 (2015), 45-54. doi: 10.3934/dcdss.2015.8.45. [6] P. Blondeau, Mechanics of coastal forms, Ann. Rev. Fluids Mech., 33 (2001), 339-370. [7] J. A. Brizard, Nonlinear gyrokinetic Vlasov equation for toroidally rotating axisymmetric tokamaks, Physics of Plasmas, 2 (1995), 459-471. doi: 10.1063/1.871465. [8] N. Crouseilles, E Frenod, S. Hirstoaga and A. Mouton, Two-Scale Macro-Micro decomposition of the Vlasov equation with a strong magnetic field, Mathematical Models and Methods in Applied Sciences, 23 (2013), 1527-1559. doi: 10.1142/S0218202513500152. [9] H. J. De Vriend, Steady Flow in Shallow Channel Bends, PhD thesis, Delft Univ. of Technology, 1981. [10] D. H. E. Dubin, J. A. Krommes, C. Oberman and W. W. Lee, Nonlinear gyrokinetic equations, Physics of Fluids, 26 (1983), 3524-3535. doi: 10.1063/1.864113. [11] I. Faye, E. Frénod and D. Seck, Singularly perturbed degenerated parabolic equations and application to seabed morphodynamics in tided environment, Discrete and Continuous Dynamical Systems - Serie A, 29 (2011), 1001-1030. doi: 10.3934/dcds.2011.29.1001. [12] I. Faye, E. Frénod and D. Seck, Two-scale numerical simulation of sand transport problems, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, 8 (2015), 151-168. doi: 10.3934/dcdss.2015.8.151. [13] I. Faye, E. Frénod and D. Seck, Long term behaviour of singularly perturbed parabolic degenerated equation, Journal of Nonlinear Analysis and Application, In press. [14] B. W. Flemming, The role of grain size, water depth and flow velocity as scaling factors controlling the size of subaqueous dunes, In A. Trentesaux and T. Garlan, editors, Marine Sandwave Dynamics, International Workshop, March 23-24 2000. University of Lille 1, France, 2000. [15] E. Frenod, A PDE-like toy-model of territory working, Submitted. [16] E. Frenod, E. Hirstoaga and S. Sonnendrücker, An exponential integrator for a highly oscillatory Vlasov equation, Discrete and Continuous Dynamical Systems - Series S, 8 (2015), 169-183. doi: 10.3934/dcdss.2015.8.169. [17] E. Frénod, S. Hirstoaga, M. Lutz and E. Sonnendrücker, Long time behaviour of an exponential integrator for a vlasov-poisson system with strong magnetic field, Communication in Computational Physics, 18 (2015), 263-296. doi: 10.4208/cicp.070214.160115a. [18] E. Frénod and M. Lutz, The Gyro-Kinetic Approximation: An attempt at explaining the method based on Darboux Algorithm and Lie Transform, Proceeding of Inria Fusion Summer School, September 2011, JLLL, UPMC. [19] E. Frénod and M. Lutz, On the geometrical gyro-kinetic theory, Kinetic and Related Models, 7 (2014), 621-659. doi: 10.3934/krm.2014.7.621. [20] E. Frénod, A. Mouton and E. Sonnendrücker, Two scale numerical simulation of the weakly compressible 1d isentropic Euler equations, Nümerishe Mathmatik, 108 (2007), 263-293. doi: 10.1007/s00211-007-0116-8. [21] E. Frénod, F. Salvarani and E. Sonnendrücker, Long time simulation of a beam in a periodic focusing channel via a two-scale PIC-method, Mathematical Models and Methods in Applied Sciences, 19 (2009), 175-197. doi: 10.1142/S0218202509003395. [22] E. A. Frieman and L. Chen, Nonlinear gyrokinetic equations for low-frequency electromagnetic waves in general plasma equilibria, Physics of Fluids, 25 (1982), 502-508. doi: 10.1063/1.863762. [23] P. E. Gadd, W. Lavelle and D. J. P. Swift, Estimates of sand transport on the New York shelf using near-bottom current meter observations, J. Sed. Petrol., 48 (1978), 239-252. [24] C. S. Gardner, Adiabatic invariants of periodic classical systems, Physical Rieview, 115 (1959), 791-794. doi: 10.1103/PhysRev.115.791. [25] V. Grandgirard, M. Brunetti, P. Bertrand, N. Besse, X. Garbet, P. Ghendrih, G. Manfredi, Y. Sarazin, O. Sauter, E. Sonnendrücker, J. Vaclavik and L. Villard, A drift-kinetic semi-lagrangian 4d code for ion turbulence simulation, Journal of Computational Physics, 217 (2006), 395-423. doi: 10.1016/j.jcp.2006.01.023. [26] V. Grandgirard, Y. Sarazin, P Angelino, A. Bottino, N. Crouseilles, G. Darmet, G. Dif-Pradalier, X. Garbet, Ph. Ghendrih, S. Jolliet, G. Latu, E. Sonnendrücker and L. Villard, Global full-$f$ gyrokinetic simulations of plasma turbulence, Plasma Physics and Controlled Fusion, 49 (2007), B173. [27] T. S. Hahm, Nonlinear gyrokinetic equations for tokamak microturbulence, Physics of Fluids, 31 (1988), 2670-2673. doi: 10.1063/1.866544. [28] T. S. Hahm, W. W. Lee and A. Brizard, Nonlinear gyrokinetic theory for finite-beta plasmas, Physics of Fluids, 31 (1988), 1940-1948. doi: 10.1063/1.866641. [29] P. Henning and M. Ohlberger, The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift, Networks and Heterogeneous Media (NHM), 5 (2010), 711-744. doi: 10.3934/nhm.2010.5.711. [30] P. Henning and M. Ohlberger, Error control and adaptivity for heterogeneous multiscale approximations of nonlinear monotone problems, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, 8 (2015), 119-150. doi: 10.3934/dcdss.2015.8.119. [31] D. Idier, D. Astruc and S. J. M. H. Hulcher, Influence of bed roughness on dune and megaripple generation, Geophysical Research Letters, 31 (2004), 1-5. doi: 10.1029/2004GL019969. [32] B. Johns, R. Soulsby and T. Chesher, The modelling of sand waves evolution resulting from suspended and bed load transport of sediment, J. Hydraul. Reseach, 28 (1990), 355-374. [33] J. Kennedy, The formation of sediment ripples, dunes and antidunes, Ann. Rev. Fluids Mech., 1 (1969), 147-168. doi: 10.1146/annurev.fl.01.010169.001051. [34] P.-V. Koseleff, Comparison between deprit and dragt-finn perturbation methods, Celestial Mech. Dynam. Astronom., 58 (1994), 17-36. doi: 10.1007/BF00692115. [35] M. D. Kruskal, Plasma Physics, chapter Elementary Orbit and Drift Theory, International Atomic Energy Agency, Vienna, 1965. [36] V. Laptev, Deterministic homogenization for media with barriers, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, 8 (2015), 29-44. doi: 10.3934/dcdss.2015.8.29. [37] F. Legoll and W. Minvielle, Variance reduction using antithetic variables for a nonlinear convex stochastic homogenization problem, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, 8 (2015), 1-27. doi: 10.3934/dcdss.2015.8.1. [38] R. G. Littlejohn, A guiding center Hamiltonian: A new approach, Journal of Mathematical Physics, 20 (1979), 2445-2458. doi: 10.1063/1.524053. [39] R. G. Littlejohn, Hamiltonian formulation of guiding center motion, Physics of Fluids, 24 (1981), 1730-1749. doi: 10.1063/1.863594. [40] R. G. Littlejohn, Hamiltonian perturbation theory in noncanonical coordinates, Journal of Mathematical Physics, 23 (1982), 742-747. doi: 10.1063/1.525429. [41] M. Lutz, Application of Lie transform techniques for simulation of a charged particle beam, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, 8 (2015), 185-221. doi: 10.3934/dcdss.2015.8.185. [42] E. Meyer-Peter and R. Müller, Formulas for bed-load transport, The Second Meeting of the International Association for Hydraulic Structures, Appendix 2, (1948), 39-44. [43] T. G. Northrop, The guiding center approximation to charged particle motion, Annals of Physics, 15 (1961), 79-101. doi: 10.1016/0003-4916(61)90167-1. [44] T. G. Northrop and J. A. Rome, Extensions of guiding center motion to higher order, Physics of Fluids, 21 (1978), 384-389. doi: 10.1063/1.862226. [45] F. I. Parra and P. J. Catto, Limitations of gyrokinetics on transport time scales, Plasma Physics and Controlled Fusion, 50 (2008), 065014. doi: 10.1088/0741-3335/50/6/065014. [46] F. I. Parra and P. J. Catto, Gyrokinetic equivalence, Plasma Physics and Controlled Fusion, 51 (2009), 065002. doi: 10.1088/0741-3335/51/6/065002. [47] F. I. Parra and P. J. Catto, Turbulent transport of toroidal angular momentum in low flow gyrokinetics, Plasma Physics and Controlled Fusion, 52 (2010), 045004. [48] Tartar, Multi-scales h-measures, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, 8 (2015), 77-90. doi: 10.3934/dcdss.2015.8.77. [49] L. C. Van Rijn, Handbook on Sediment Transport by Current and Waves, Technical Report H461:12.1-12.27, Delft Hydraulics, 1989. [50] X. Xu and S. Yue, Homogenization of thermal-hydro-mass transfer processes, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, 8 (2015), 55-76. doi: 10.3934/dcdss.2015.8.55.

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References:
 [1] A. Abdulle, Y. Bai and G. Vilmart, Reduced basis finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, 8 (2015), 91-118. doi: 10.3934/dcdss.2015.8.91. [2] P. Ailliot, E. Frénod and V. Monbet, Long term object drift forecast in the ocean with tide and wind, Multiscale Modeling and Simulations, 5 (2006), 514-531. doi: 10.1137/050639727. [3] G. Allaire and R. Brizzi, A multiscale finite element method for numerical homogenization, SIAM Multiscale Modeling and Simulations, 4 (2005), 790-812. doi: 10.1137/040611239. [4] A. Back and E. Frénod, Geometric two-scale convergence on manifold and applications to the Vlasov equation, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, 8 (2015), 223-241. doi: 10.3934/dcdss.2015.8.223. [5] J.-P. Bernard, E. Frénod and A. Rousseau, Paralic confinement computations in coastal environment with interlocked areas, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, 8 (2015), 45-54. doi: 10.3934/dcdss.2015.8.45. [6] P. Blondeau, Mechanics of coastal forms, Ann. Rev. Fluids Mech., 33 (2001), 339-370. [7] J. A. Brizard, Nonlinear gyrokinetic Vlasov equation for toroidally rotating axisymmetric tokamaks, Physics of Plasmas, 2 (1995), 459-471. doi: 10.1063/1.871465. [8] N. Crouseilles, E Frenod, S. Hirstoaga and A. Mouton, Two-Scale Macro-Micro decomposition of the Vlasov equation with a strong magnetic field, Mathematical Models and Methods in Applied Sciences, 23 (2013), 1527-1559. doi: 10.1142/S0218202513500152. [9] H. J. De Vriend, Steady Flow in Shallow Channel Bends, PhD thesis, Delft Univ. of Technology, 1981. [10] D. H. E. Dubin, J. A. Krommes, C. Oberman and W. W. Lee, Nonlinear gyrokinetic equations, Physics of Fluids, 26 (1983), 3524-3535. doi: 10.1063/1.864113. [11] I. Faye, E. Frénod and D. Seck, Singularly perturbed degenerated parabolic equations and application to seabed morphodynamics in tided environment, Discrete and Continuous Dynamical Systems - Serie A, 29 (2011), 1001-1030. doi: 10.3934/dcds.2011.29.1001. [12] I. Faye, E. Frénod and D. Seck, Two-scale numerical simulation of sand transport problems, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, 8 (2015), 151-168. doi: 10.3934/dcdss.2015.8.151. [13] I. Faye, E. Frénod and D. Seck, Long term behaviour of singularly perturbed parabolic degenerated equation, Journal of Nonlinear Analysis and Application, In press. [14] B. W. Flemming, The role of grain size, water depth and flow velocity as scaling factors controlling the size of subaqueous dunes, In A. Trentesaux and T. Garlan, editors, Marine Sandwave Dynamics, International Workshop, March 23-24 2000. University of Lille 1, France, 2000. [15] E. Frenod, A PDE-like toy-model of territory working, Submitted. [16] E. Frenod, E. Hirstoaga and S. Sonnendrücker, An exponential integrator for a highly oscillatory Vlasov equation, Discrete and Continuous Dynamical Systems - Series S, 8 (2015), 169-183. doi: 10.3934/dcdss.2015.8.169. [17] E. Frénod, S. Hirstoaga, M. Lutz and E. Sonnendrücker, Long time behaviour of an exponential integrator for a vlasov-poisson system with strong magnetic field, Communication in Computational Physics, 18 (2015), 263-296. doi: 10.4208/cicp.070214.160115a. [18] E. Frénod and M. Lutz, The Gyro-Kinetic Approximation: An attempt at explaining the method based on Darboux Algorithm and Lie Transform, Proceeding of Inria Fusion Summer School, September 2011, JLLL, UPMC. [19] E. Frénod and M. Lutz, On the geometrical gyro-kinetic theory, Kinetic and Related Models, 7 (2014), 621-659. doi: 10.3934/krm.2014.7.621. [20] E. Frénod, A. Mouton and E. Sonnendrücker, Two scale numerical simulation of the weakly compressible 1d isentropic Euler equations, Nümerishe Mathmatik, 108 (2007), 263-293. doi: 10.1007/s00211-007-0116-8. [21] E. Frénod, F. Salvarani and E. Sonnendrücker, Long time simulation of a beam in a periodic focusing channel via a two-scale PIC-method, Mathematical Models and Methods in Applied Sciences, 19 (2009), 175-197. doi: 10.1142/S0218202509003395. [22] E. A. Frieman and L. Chen, Nonlinear gyrokinetic equations for low-frequency electromagnetic waves in general plasma equilibria, Physics of Fluids, 25 (1982), 502-508. doi: 10.1063/1.863762. [23] P. E. Gadd, W. Lavelle and D. J. P. Swift, Estimates of sand transport on the New York shelf using near-bottom current meter observations, J. Sed. Petrol., 48 (1978), 239-252. [24] C. S. Gardner, Adiabatic invariants of periodic classical systems, Physical Rieview, 115 (1959), 791-794. doi: 10.1103/PhysRev.115.791. [25] V. Grandgirard, M. Brunetti, P. Bertrand, N. Besse, X. Garbet, P. Ghendrih, G. Manfredi, Y. Sarazin, O. Sauter, E. Sonnendrücker, J. Vaclavik and L. Villard, A drift-kinetic semi-lagrangian 4d code for ion turbulence simulation, Journal of Computational Physics, 217 (2006), 395-423. doi: 10.1016/j.jcp.2006.01.023. [26] V. Grandgirard, Y. Sarazin, P Angelino, A. Bottino, N. Crouseilles, G. Darmet, G. Dif-Pradalier, X. Garbet, Ph. Ghendrih, S. Jolliet, G. Latu, E. Sonnendrücker and L. Villard, Global full-$f$ gyrokinetic simulations of plasma turbulence, Plasma Physics and Controlled Fusion, 49 (2007), B173. [27] T. S. Hahm, Nonlinear gyrokinetic equations for tokamak microturbulence, Physics of Fluids, 31 (1988), 2670-2673. doi: 10.1063/1.866544. [28] T. S. Hahm, W. W. Lee and A. Brizard, Nonlinear gyrokinetic theory for finite-beta plasmas, Physics of Fluids, 31 (1988), 1940-1948. doi: 10.1063/1.866641. [29] P. Henning and M. Ohlberger, The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift, Networks and Heterogeneous Media (NHM), 5 (2010), 711-744. doi: 10.3934/nhm.2010.5.711. [30] P. Henning and M. Ohlberger, Error control and adaptivity for heterogeneous multiscale approximations of nonlinear monotone problems, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, 8 (2015), 119-150. doi: 10.3934/dcdss.2015.8.119. [31] D. Idier, D. Astruc and S. J. M. H. Hulcher, Influence of bed roughness on dune and megaripple generation, Geophysical Research Letters, 31 (2004), 1-5. doi: 10.1029/2004GL019969. [32] B. Johns, R. Soulsby and T. Chesher, The modelling of sand waves evolution resulting from suspended and bed load transport of sediment, J. Hydraul. Reseach, 28 (1990), 355-374. [33] J. Kennedy, The formation of sediment ripples, dunes and antidunes, Ann. Rev. Fluids Mech., 1 (1969), 147-168. doi: 10.1146/annurev.fl.01.010169.001051. [34] P.-V. Koseleff, Comparison between deprit and dragt-finn perturbation methods, Celestial Mech. Dynam. Astronom., 58 (1994), 17-36. doi: 10.1007/BF00692115. [35] M. D. Kruskal, Plasma Physics, chapter Elementary Orbit and Drift Theory, International Atomic Energy Agency, Vienna, 1965. [36] V. Laptev, Deterministic homogenization for media with barriers, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, 8 (2015), 29-44. doi: 10.3934/dcdss.2015.8.29. [37] F. Legoll and W. Minvielle, Variance reduction using antithetic variables for a nonlinear convex stochastic homogenization problem, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, 8 (2015), 1-27. doi: 10.3934/dcdss.2015.8.1. [38] R. G. Littlejohn, A guiding center Hamiltonian: A new approach, Journal of Mathematical Physics, 20 (1979), 2445-2458. doi: 10.1063/1.524053. [39] R. G. Littlejohn, Hamiltonian formulation of guiding center motion, Physics of Fluids, 24 (1981), 1730-1749. doi: 10.1063/1.863594. [40] R. G. Littlejohn, Hamiltonian perturbation theory in noncanonical coordinates, Journal of Mathematical Physics, 23 (1982), 742-747. doi: 10.1063/1.525429. [41] M. Lutz, Application of Lie transform techniques for simulation of a charged particle beam, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, 8 (2015), 185-221. doi: 10.3934/dcdss.2015.8.185. [42] E. Meyer-Peter and R. Müller, Formulas for bed-load transport, The Second Meeting of the International Association for Hydraulic Structures, Appendix 2, (1948), 39-44. [43] T. G. Northrop, The guiding center approximation to charged particle motion, Annals of Physics, 15 (1961), 79-101. doi: 10.1016/0003-4916(61)90167-1. [44] T. G. Northrop and J. A. Rome, Extensions of guiding center motion to higher order, Physics of Fluids, 21 (1978), 384-389. doi: 10.1063/1.862226. [45] F. I. Parra and P. J. Catto, Limitations of gyrokinetics on transport time scales, Plasma Physics and Controlled Fusion, 50 (2008), 065014. doi: 10.1088/0741-3335/50/6/065014. [46] F. I. Parra and P. J. Catto, Gyrokinetic equivalence, Plasma Physics and Controlled Fusion, 51 (2009), 065002. doi: 10.1088/0741-3335/51/6/065002. [47] F. I. Parra and P. J. Catto, Turbulent transport of toroidal angular momentum in low flow gyrokinetics, Plasma Physics and Controlled Fusion, 52 (2010), 045004. [48] Tartar, Multi-scales h-measures, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, 8 (2015), 77-90. doi: 10.3934/dcdss.2015.8.77. [49] L. C. Van Rijn, Handbook on Sediment Transport by Current and Waves, Technical Report H461:12.1-12.27, Delft Hydraulics, 1989. [50] X. Xu and S. Yue, Homogenization of thermal-hydro-mass transfer processes, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, 8 (2015), 55-76. doi: 10.3934/dcdss.2015.8.55.
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