# 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 & 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.  doi: 10.3934/dcdss.2015.8.91.  Google Scholar [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.  doi: 10.1137/050639727.  Google Scholar [3] G. Allaire and R. Brizzi, A multiscale finite element method for numerical homogenization,, SIAM Multiscale Modeling and Simulations, 4 (2005), 790.  doi: 10.1137/040611239.  Google Scholar [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.  doi: 10.3934/dcdss.2015.8.223.  Google Scholar [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.  doi: 10.3934/dcdss.2015.8.45.  Google Scholar [6] P. Blondeau, Mechanics of coastal forms,, Ann. Rev. Fluids Mech., 33 (2001), 339.   Google Scholar [7] J. A. Brizard, Nonlinear gyrokinetic Vlasov equation for toroidally rotating axisymmetric tokamaks,, Physics of Plasmas, 2 (1995), 459.  doi: 10.1063/1.871465.  Google Scholar [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.  doi: 10.1142/S0218202513500152.  Google Scholar [9] H. J. De Vriend, Steady Flow in Shallow Channel Bends,, PhD thesis, (1981).   Google Scholar [10] D. H. E. Dubin, J. A. Krommes, C. Oberman and W. W. Lee, Nonlinear gyrokinetic equations,, Physics of Fluids, 26 (1983), 3524.  doi: 10.1063/1.864113.  Google Scholar [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.  doi: 10.3934/dcds.2011.29.1001.  Google Scholar [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.  doi: 10.3934/dcdss.2015.8.151.  Google Scholar [13] I. Faye, E. Frénod and D. Seck, Long term behaviour of singularly perturbed parabolic degenerated equation,, Journal of Nonlinear Analysis and Application, ().   Google Scholar [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, (2000), 23.   Google Scholar [15] E. Frenod, A PDE-like toy-model of territory working,, Submitted., ().   Google Scholar [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.  doi: 10.3934/dcdss.2015.8.169.  Google Scholar [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.  doi: 10.4208/cicp.070214.160115a.  Google Scholar [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, (2011).   Google Scholar [19] E. Frénod and M. Lutz, On the geometrical gyro-kinetic theory,, Kinetic and Related Models, 7 (2014), 621.  doi: 10.3934/krm.2014.7.621.  Google Scholar [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.  doi: 10.1007/s00211-007-0116-8.  Google Scholar [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.  doi: 10.1142/S0218202509003395.  Google Scholar [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.  doi: 10.1063/1.863762.  Google Scholar [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.   Google Scholar [24] C. S. Gardner, Adiabatic invariants of periodic classical systems,, Physical Rieview, 115 (1959), 791.  doi: 10.1103/PhysRev.115.791.  Google Scholar [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.  doi: 10.1016/j.jcp.2006.01.023.  Google Scholar [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).   Google Scholar [27] T. S. Hahm, Nonlinear gyrokinetic equations for tokamak microturbulence,, Physics of Fluids, 31 (1988), 2670.  doi: 10.1063/1.866544.  Google Scholar [28] T. S. Hahm, W. W. Lee and A. Brizard, Nonlinear gyrokinetic theory for finite-beta plasmas,, Physics of Fluids, 31 (1988), 1940.  doi: 10.1063/1.866641.  Google Scholar [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.  doi: 10.3934/nhm.2010.5.711.  Google Scholar [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.  doi: 10.3934/dcdss.2015.8.119.  Google Scholar [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.  doi: 10.1029/2004GL019969.  Google Scholar [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.   Google Scholar [33] J. Kennedy, The formation of sediment ripples, dunes and antidunes,, Ann. Rev. Fluids Mech., 1 (1969), 147.  doi: 10.1146/annurev.fl.01.010169.001051.  Google Scholar [34] P.-V. Koseleff, Comparison between deprit and dragt-finn perturbation methods,, Celestial Mech. Dynam. Astronom., 58 (1994), 17.  doi: 10.1007/BF00692115.  Google Scholar [35] M. D. Kruskal, Plasma Physics, chapter Elementary Orbit and Drift Theory,, International Atomic Energy Agency, (1965).   Google Scholar [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.  doi: 10.3934/dcdss.2015.8.29.  Google Scholar [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.  doi: 10.3934/dcdss.2015.8.1.  Google Scholar [38] R. G. Littlejohn, A guiding center Hamiltonian: A new approach,, Journal of Mathematical Physics, 20 (1979), 2445.  doi: 10.1063/1.524053.  Google Scholar [39] R. G. Littlejohn, Hamiltonian formulation of guiding center motion,, Physics of Fluids, 24 (1981), 1730.  doi: 10.1063/1.863594.  Google Scholar [40] R. G. Littlejohn, Hamiltonian perturbation theory in noncanonical coordinates,, Journal of Mathematical Physics, 23 (1982), 742.  doi: 10.1063/1.525429.  Google Scholar [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.  doi: 10.3934/dcdss.2015.8.185.  Google Scholar [42] E. Meyer-Peter and R. Müller, Formulas for bed-load transport,, The Second Meeting of the International Association for Hydraulic Structures, (1948), 39.   Google Scholar [43] T. G. Northrop, The guiding center approximation to charged particle motion,, Annals of Physics, 15 (1961), 79.  doi: 10.1016/0003-4916(61)90167-1.  Google Scholar [44] T. G. Northrop and J. A. Rome, Extensions of guiding center motion to higher order,, Physics of Fluids, 21 (1978), 384.  doi: 10.1063/1.862226.  Google Scholar [45] F. I. Parra and P. J. Catto, Limitations of gyrokinetics on transport time scales,, Plasma Physics and Controlled Fusion, 50 (2008).  doi: 10.1088/0741-3335/50/6/065014.  Google Scholar [46] F. I. Parra and P. J. Catto, Gyrokinetic equivalence,, Plasma Physics and Controlled Fusion, 51 (2009).  doi: 10.1088/0741-3335/51/6/065002.  Google Scholar [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).   Google Scholar [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.  doi: 10.3934/dcdss.2015.8.77.  Google Scholar [49] L. C. Van Rijn, Handbook on Sediment Transport by Current and Waves,, Technical Report H461:12.1-12.27, (1989), 1.   Google Scholar [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.  doi: 10.3934/dcdss.2015.8.55.  Google Scholar

show all references

##### 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.  doi: 10.3934/dcdss.2015.8.91.  Google Scholar [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.  doi: 10.1137/050639727.  Google Scholar [3] G. Allaire and R. Brizzi, A multiscale finite element method for numerical homogenization,, SIAM Multiscale Modeling and Simulations, 4 (2005), 790.  doi: 10.1137/040611239.  Google Scholar [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.  doi: 10.3934/dcdss.2015.8.223.  Google Scholar [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.  doi: 10.3934/dcdss.2015.8.45.  Google Scholar [6] P. Blondeau, Mechanics of coastal forms,, Ann. Rev. Fluids Mech., 33 (2001), 339.   Google Scholar [7] J. A. Brizard, Nonlinear gyrokinetic Vlasov equation for toroidally rotating axisymmetric tokamaks,, Physics of Plasmas, 2 (1995), 459.  doi: 10.1063/1.871465.  Google Scholar [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.  doi: 10.1142/S0218202513500152.  Google Scholar [9] H. J. De Vriend, Steady Flow in Shallow Channel Bends,, PhD thesis, (1981).   Google Scholar [10] D. H. E. Dubin, J. A. Krommes, C. Oberman and W. W. Lee, Nonlinear gyrokinetic equations,, Physics of Fluids, 26 (1983), 3524.  doi: 10.1063/1.864113.  Google Scholar [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.  doi: 10.3934/dcds.2011.29.1001.  Google Scholar [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.  doi: 10.3934/dcdss.2015.8.151.  Google Scholar [13] I. Faye, E. Frénod and D. Seck, Long term behaviour of singularly perturbed parabolic degenerated equation,, Journal of Nonlinear Analysis and Application, ().   Google Scholar [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, (2000), 23.   Google Scholar [15] E. Frenod, A PDE-like toy-model of territory working,, Submitted., ().   Google Scholar [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.  doi: 10.3934/dcdss.2015.8.169.  Google Scholar [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.  doi: 10.4208/cicp.070214.160115a.  Google Scholar [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, (2011).   Google Scholar [19] E. Frénod and M. Lutz, On the geometrical gyro-kinetic theory,, Kinetic and Related Models, 7 (2014), 621.  doi: 10.3934/krm.2014.7.621.  Google Scholar [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.  doi: 10.1007/s00211-007-0116-8.  Google Scholar [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.  doi: 10.1142/S0218202509003395.  Google Scholar [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.  doi: 10.1063/1.863762.  Google Scholar [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.   Google Scholar [24] C. S. Gardner, Adiabatic invariants of periodic classical systems,, Physical Rieview, 115 (1959), 791.  doi: 10.1103/PhysRev.115.791.  Google Scholar [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.  doi: 10.1016/j.jcp.2006.01.023.  Google Scholar [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).   Google Scholar [27] T. S. Hahm, Nonlinear gyrokinetic equations for tokamak microturbulence,, Physics of Fluids, 31 (1988), 2670.  doi: 10.1063/1.866544.  Google Scholar [28] T. S. Hahm, W. W. Lee and A. Brizard, Nonlinear gyrokinetic theory for finite-beta plasmas,, Physics of Fluids, 31 (1988), 1940.  doi: 10.1063/1.866641.  Google Scholar [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.  doi: 10.3934/nhm.2010.5.711.  Google Scholar [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.  doi: 10.3934/dcdss.2015.8.119.  Google Scholar [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.  doi: 10.1029/2004GL019969.  Google Scholar [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.   Google Scholar [33] J. Kennedy, The formation of sediment ripples, dunes and antidunes,, Ann. Rev. Fluids Mech., 1 (1969), 147.  doi: 10.1146/annurev.fl.01.010169.001051.  Google Scholar [34] P.-V. Koseleff, Comparison between deprit and dragt-finn perturbation methods,, Celestial Mech. Dynam. Astronom., 58 (1994), 17.  doi: 10.1007/BF00692115.  Google Scholar [35] M. D. Kruskal, Plasma Physics, chapter Elementary Orbit and Drift Theory,, International Atomic Energy Agency, (1965).   Google Scholar [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.  doi: 10.3934/dcdss.2015.8.29.  Google Scholar [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.  doi: 10.3934/dcdss.2015.8.1.  Google Scholar [38] R. G. Littlejohn, A guiding center Hamiltonian: A new approach,, Journal of Mathematical Physics, 20 (1979), 2445.  doi: 10.1063/1.524053.  Google Scholar [39] R. G. Littlejohn, Hamiltonian formulation of guiding center motion,, Physics of Fluids, 24 (1981), 1730.  doi: 10.1063/1.863594.  Google Scholar [40] R. G. Littlejohn, Hamiltonian perturbation theory in noncanonical coordinates,, Journal of Mathematical Physics, 23 (1982), 742.  doi: 10.1063/1.525429.  Google Scholar [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.  doi: 10.3934/dcdss.2015.8.185.  Google Scholar [42] E. Meyer-Peter and R. 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