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. 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

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