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Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative
Damping, stabilization, and numerical filtering for the modeling and the simulation of time dependent PDEs
Laboratoire LAMFA (UMR CNRS 7352), Université de Picardie Jules Verne, 33 rue Saint Leu, 80039 Amiens Cédex, France |
We present here different situations in which the filtering of high or low modes is used either for stabilizing semi-implicit numerical schemes when solving nonlinear parabolic equations, or for building adapted damping operators in the case of dispersive equation. We consider numerical filtering provided by mutigrid-like techniques as well as the filtering resulting from operator with monotone symbols. Our approach applies to several discretization techniques and we focus on finite elements and finite differences. Numerical illustrations are given on Cahn-Hilliard, Korteweig-de Vries and Kuramoto-Sivashinsky equations.
References:
[1] |
H. Abboud, C. Al Kosseifi and J.-P. Chehab, A stabilized bi-grid method for Allen-Cahn equation in finite elements, Comput. Appl. Math., 38 (2019), Paper No. 35, 27 pp.
doi: 10.1007/s40314-019-0781-0. |
[2] |
M. Abounouh, H. Al Moatassime, J-P. Chehab, S. Dumont and O. Goubet,
Discrete Schrödinger equations and dissipative dynamical systems, Commun. Pure Appl. Anal., 7 (2008), 211-227.
doi: 10.3934/cpaa.2008.7.211. |
[3] |
K. Adamy, A. Bousquet, S. Faure, J. Laminie and R. Temam,
A multilevel method for finite volume discretization of the two-dimensional nonlinear Shallow-Water equations, Ocean Modelling, 33 (2010), 235-256.
doi: 10.1016/j.ocemod.2010.02.006. |
[4] |
R. E. Bank, Hierarchical bases and the finite element method, (English) Iserles, A. (ed.), Acta Numerica, Vol. 5, 1996. Cambridge: Cambridge University Press. (1996), 1-43.
doi: 10.1017/S0962492900002610. |
[5] |
J. Bona and R. Smith, Existence of solutions to the Korteweg-de Vries initial value problem, In Nonlinear Wave Motion (Proc. AMS-SIAM Summer Sem., Clarkson Coll. Tech., Potsdam, N.Y., 1972), Lectures in Appl. Math., Amer. Math. Soc., Providence, R.I., 15 (1974), 179-180. |
[6] |
J. L. Bona, M. Chen and J.-C. Saut,
Boussinesq equations and other systems for small amplitude long waves in nonlinear dispersive media: Ⅱ. the nonlinear theory, Nonlinearity, 17 (2004), 925-952.
doi: 10.1088/0951-7715/17/3/010. |
[7] |
J. L. Bona and R. Smith,
The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A, 278 (1975), 555-601.
doi: 10.1098/rsta.1975.0035. |
[8] |
M. Brachet and J.-P. Chehab,
Stabilized times schemes for high accurate finite differences solutions of nonlinear parabolic equations, J. Sci. Comput., 69 (2016), 946-982.
doi: 10.1007/s10915-016-0223-8. |
[9] |
M. Brachet and J.-P. Chehab,
Fast and stable schemes for phase fields models, Comput. Math. Appl., 80 (2020), 1683-1713.
doi: 10.1016/j.camwa.2020.07.015. |
[10] |
M. Brachet, Schémas Compacts Hermitiens sur la sphère - Applications en Climatologie et Océanographie Numérique, Thèse, Université de Lorraine, Jully 2018 (in French). Google Scholar |
[11] |
C. Brezinski and J.-P. Chehab,
Nonlinear hybrid procedures and fixed point iterations, Numer. Funct. Anal. Optim., 19 (1998), 465-487.
doi: 10.1080/01630569808816839. |
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M. Cabral and R. Rosa,
Chaos for a damped and forced KdV equation, Phys. D, 192 (2004), 265-278.
doi: 10.1016/j.physd.2004.01.023. |
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C. Calgaro, J.-P. Chehab, J. Laminie and E. Zahrouni, Séparation des échelles et schémas multiniveaux pour les équations d'ondes non-linéaires, (French) [Scale separation and multilevel schemes for nonlinear wave equations] CANUM, (2008), 180-208, ESAIM Proc., 27, EDP Sci., Les Ulis, 2009.
doi: 10.1051/proc/2009027. |
[14] |
C. Calgaro, A. Debussche and J. Laminie,
On a multilevel approach for the two-dimensional Navier-Stokes equations with finite elements, Finite Elements in Fluids. Internat. J. Numer. Methods Fluids, 27 (1998), 241-258.
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Dynamical multilevel schemes for the solution of evolution equations by hierarchical finite element discretization, Appl. Numer. Math., 23 (1997), 403-442.
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J.-P. Chehab and B. Costa,
Multiparameter methods for evolutionary equations, Numerical Algorithms, 34 (2003), 245-257.
doi: 10.1023/B:NUMA.0000005401.91113.1f. |
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J.-P. Chehab and B. Costa, Multiparameter Extensions of Iterative Processes Rapport Technique du Laboratoire de Mathmatiques d'Orsay, 2002. Google Scholar |
[18] |
J.-P. Chehab and B. Costa,
Time explicit schemes and spatial finite differences splittings, J. Sci. Comput., 20 (2004), 159-189.
doi: 10.1023/B:JOMP.0000008719.48134.4f. |
[19] |
J.-P. Chehab, P. Garnier and Y. Mammeri,
Long-time behavior of solutions of a BBM equation with generalized damping, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1897-1915.
doi: 10.3934/dcdsb.2015.20.1897. |
[20] |
J.-P. Chehab, P. Garnier and Y. Mammeri, Numerical solution of the generalized Kadomtsev-Petviashvili equations with compact finite difference schemes, submitted. Google Scholar |
[21] |
J.-P. Chehab and G. Sadaka,
Numerical study of a family of dissipative KdV equations, Commun. Pure Appl. Anal., 12 (2013), 519-546.
doi: 10.3934/cpaa.2013.12.519. |
[22] |
J.-P. Chehab and G. Sadaka,
On damping rates of dissipative KdV equations, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 1487-1506.
doi: 10.3934/dcdss.2013.6.1487. |
[23] |
M. Chen, S. Dumont, L. Dupaigne and O. Goubet,
Decay of solutions to a water wave model with nonlocal viscous dispersive term, Discrete Contin. Dyn. Syst., 27 (2010), 1473-1492.
doi: 10.3934/dcds.2010.27.1473. |
[24] |
A. Cohen, Numerical Analysis of Wavelet Methods, North-Holland Publishing Co., Amsterdam, 2003. |
[25] |
B. Costa, L. Dettori, D. Gottlieb and R. Temam., Time marching techniques for the nonlinear Galerkin method, SIAM J. SC. Comp., 23 (2001), 46-65. Google Scholar |
[26] |
A. Debussche, J. Laminie and E. Zahrouni,
A dynamical multi-level scheme for the Burgers equation: Wavelet and hierarchical finite element, J. Sci. Comput., 25 (2005), 445-497.
doi: 10.1007/s10915-004-4806-4. |
[27] |
T. Dubois, F. Jauberteau and R. Temam., Dynamic Multilevel Methods and the Numerical Simulation of Turbulence., Cambridge University Press, Cambridge, 1999.
![]() |
[28] |
T. Dubois, F. Jauberteau and R. Temam,
Incremental unknowns, multilevel methods and the numerical simulation of turbulence, Comput. Methods Appl. Mech. Engrg., 159 (1998), 123-189.
doi: 10.1016/S0045-7825(98)80106-0. |
[29] |
S. Dumont and J.-B. Duval,
Numerical investigation of asymptotical properties of solutions to models for water waves with non local viscosity, Int. J. Numer. Anal. Model., 10 (2013), 333-349.
|
[30] |
S. Dumont and I. Manoubi,
Numerical analysis of a water wave model with a nonlocal viscous dispersive term using the diffusive approach, Math. Methods Appl. Sci., 41 (2018), 4810-4826.
|
[31] |
D. Dutykh,
Viscous-potential free-surface flows and long wave modelling, Eur. J. Mech. B Fluids, 28 (2009), 430-443.
doi: 10.1016/j.euromechflu.2008.11.003. |
[32] |
D. Dutykh and F. Dias,
Viscous potentiel free-surface flows in a fluid layer of finite depth, C. R. Math. Acad. Sci. Paris, 345 (2007), 113-118.
doi: 10.1016/j.crma.2007.06.007. |
[33] |
H. Emmerich, The Diffuse Interface Approach in Materials Science Thermodynamic, Concepts and Applications of Phase-Field Models. Lecture Notes in Physics Monographs, Springer, Heidelberg, 2003. Google Scholar |
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A. Ern, J.-L. Guermond, Theory and Practice of Finite Elements, Applied Mathematical Science, 159, Springer-Verlag, New-York, 2004.
doi: 10.1007/978-1-4757-4355-5. |
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D. J. Eyre, Unconditionallly Stable One-step Scheme for Gradient Systems, June 1998, unpublished, http://www.math.utah.edu/eyre/research/methods/stable.ps. Google Scholar |
[36] |
E. Ezzoug, O. Goubet and E. Zahrouni,
Semi-discrete weakly damped nonlinear 2-D Schrödinger equation, Differential Integral Equations, 23 (2010), 237-252.
|
[37] |
S. Faure, J. Laminie and R. Temam,
Finite volume discretization and multilevel methods in flow problems, J. Sci. Comput., 25 (2005), 231-261.
doi: 10.1007/s10915-004-4642-6. |
[38] |
, FreeFem++ Page, http://www.freefem.org Google Scholar |
[39] |
P. Garnier,
Damping to prevent the blow-up of the Korteweg-de Vries equation, Commun. Pure Appl. Anal., 16 (2017), 1455-1470.
doi: 10.3934/cpaa.2017069. |
[40] |
S. Gasparin, J. Berger, D. Dutykh and N. Mendes,
Stable explicit schemes for simulation of nonlinear moisture transfer in porous materials, J. Build. Perf. Sim., 11 (2017), 129-144.
doi: 10.1080/19401493.2017.1298669. |
[41] |
J.-M. Ghidaglia,
Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time, J. Differential Equations, 74 (1988), 369-390.
doi: 10.1016/0022-0396(88)90010-1. |
[42] |
J.-M. Ghidaglia,
A note on the strong convergence towards attractors for damped forced KdV equations, J. Differential Equations, 110 (1994), 356-359.
doi: 10.1006/jdeq.1994.1071. |
[43] |
O. Goubet,
Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations, Discrete Contin. Dynam. Systems, 6 (2000), 625-644.
doi: 10.3934/dcds.2000.6.625. |
[44] |
O. Goubet and R. M. S. Rosa,
Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line, J. Differential Equations, 185 (2002), 25-53.
doi: 10.1006/jdeq.2001.4163. |
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O. Goubet and E. Zahrouni,
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show all references
References:
[1] |
H. Abboud, C. Al Kosseifi and J.-P. Chehab, A stabilized bi-grid method for Allen-Cahn equation in finite elements, Comput. Appl. Math., 38 (2019), Paper No. 35, 27 pp.
doi: 10.1007/s40314-019-0781-0. |
[2] |
M. Abounouh, H. Al Moatassime, J-P. Chehab, S. Dumont and O. Goubet,
Discrete Schrödinger equations and dissipative dynamical systems, Commun. Pure Appl. Anal., 7 (2008), 211-227.
doi: 10.3934/cpaa.2008.7.211. |
[3] |
K. Adamy, A. Bousquet, S. Faure, J. Laminie and R. Temam,
A multilevel method for finite volume discretization of the two-dimensional nonlinear Shallow-Water equations, Ocean Modelling, 33 (2010), 235-256.
doi: 10.1016/j.ocemod.2010.02.006. |
[4] |
R. E. Bank, Hierarchical bases and the finite element method, (English) Iserles, A. (ed.), Acta Numerica, Vol. 5, 1996. Cambridge: Cambridge University Press. (1996), 1-43.
doi: 10.1017/S0962492900002610. |
[5] |
J. Bona and R. Smith, Existence of solutions to the Korteweg-de Vries initial value problem, In Nonlinear Wave Motion (Proc. AMS-SIAM Summer Sem., Clarkson Coll. Tech., Potsdam, N.Y., 1972), Lectures in Appl. Math., Amer. Math. Soc., Providence, R.I., 15 (1974), 179-180. |
[6] |
J. L. Bona, M. Chen and J.-C. Saut,
Boussinesq equations and other systems for small amplitude long waves in nonlinear dispersive media: Ⅱ. the nonlinear theory, Nonlinearity, 17 (2004), 925-952.
doi: 10.1088/0951-7715/17/3/010. |
[7] |
J. L. Bona and R. Smith,
The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A, 278 (1975), 555-601.
doi: 10.1098/rsta.1975.0035. |
[8] |
M. Brachet and J.-P. Chehab,
Stabilized times schemes for high accurate finite differences solutions of nonlinear parabolic equations, J. Sci. Comput., 69 (2016), 946-982.
doi: 10.1007/s10915-016-0223-8. |
[9] |
M. Brachet and J.-P. Chehab,
Fast and stable schemes for phase fields models, Comput. Math. Appl., 80 (2020), 1683-1713.
doi: 10.1016/j.camwa.2020.07.015. |
[10] |
M. Brachet, Schémas Compacts Hermitiens sur la sphère - Applications en Climatologie et Océanographie Numérique, Thèse, Université de Lorraine, Jully 2018 (in French). Google Scholar |
[11] |
C. Brezinski and J.-P. Chehab,
Nonlinear hybrid procedures and fixed point iterations, Numer. Funct. Anal. Optim., 19 (1998), 465-487.
doi: 10.1080/01630569808816839. |
[12] |
M. Cabral and R. Rosa,
Chaos for a damped and forced KdV equation, Phys. D, 192 (2004), 265-278.
doi: 10.1016/j.physd.2004.01.023. |
[13] |
C. Calgaro, J.-P. Chehab, J. Laminie and E. Zahrouni, Séparation des échelles et schémas multiniveaux pour les équations d'ondes non-linéaires, (French) [Scale separation and multilevel schemes for nonlinear wave equations] CANUM, (2008), 180-208, ESAIM Proc., 27, EDP Sci., Les Ulis, 2009.
doi: 10.1051/proc/2009027. |
[14] |
C. Calgaro, A. Debussche and J. Laminie,
On a multilevel approach for the two-dimensional Navier-Stokes equations with finite elements, Finite Elements in Fluids. Internat. J. Numer. Methods Fluids, 27 (1998), 241-258.
doi: 10.1002/(SICI)1097-0363(199801)27:1/4<241::AID-FLD662>3.0.CO;2-4. |
[15] |
C. Calgaro, J. Laminie and R. Temam,
Dynamical multilevel schemes for the solution of evolution equations by hierarchical finite element discretization, Appl. Numer. Math., 23 (1997), 403-442.
doi: 10.1016/S0168-9274(96)00074-8. |
[16] |
J.-P. Chehab and B. Costa,
Multiparameter methods for evolutionary equations, Numerical Algorithms, 34 (2003), 245-257.
doi: 10.1023/B:NUMA.0000005401.91113.1f. |
[17] |
J.-P. Chehab and B. Costa, Multiparameter Extensions of Iterative Processes Rapport Technique du Laboratoire de Mathmatiques d'Orsay, 2002. Google Scholar |
[18] |
J.-P. Chehab and B. Costa,
Time explicit schemes and spatial finite differences splittings, J. Sci. Comput., 20 (2004), 159-189.
doi: 10.1023/B:JOMP.0000008719.48134.4f. |
[19] |
J.-P. Chehab, P. Garnier and Y. Mammeri,
Long-time behavior of solutions of a BBM equation with generalized damping, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1897-1915.
doi: 10.3934/dcdsb.2015.20.1897. |
[20] |
J.-P. Chehab, P. Garnier and Y. Mammeri, Numerical solution of the generalized Kadomtsev-Petviashvili equations with compact finite difference schemes, submitted. Google Scholar |
[21] |
J.-P. Chehab and G. Sadaka,
Numerical study of a family of dissipative KdV equations, Commun. Pure Appl. Anal., 12 (2013), 519-546.
doi: 10.3934/cpaa.2013.12.519. |
[22] |
J.-P. Chehab and G. Sadaka,
On damping rates of dissipative KdV equations, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 1487-1506.
doi: 10.3934/dcdss.2013.6.1487. |
[23] |
M. Chen, S. Dumont, L. Dupaigne and O. Goubet,
Decay of solutions to a water wave model with nonlocal viscous dispersive term, Discrete Contin. Dyn. Syst., 27 (2010), 1473-1492.
doi: 10.3934/dcds.2010.27.1473. |
[24] |
A. Cohen, Numerical Analysis of Wavelet Methods, North-Holland Publishing Co., Amsterdam, 2003. |
[25] |
B. Costa, L. Dettori, D. Gottlieb and R. Temam., Time marching techniques for the nonlinear Galerkin method, SIAM J. SC. Comp., 23 (2001), 46-65. Google Scholar |
[26] |
A. Debussche, J. Laminie and E. Zahrouni,
A dynamical multi-level scheme for the Burgers equation: Wavelet and hierarchical finite element, J. Sci. Comput., 25 (2005), 445-497.
doi: 10.1007/s10915-004-4806-4. |
[27] |
T. Dubois, F. Jauberteau and R. Temam., Dynamic Multilevel Methods and the Numerical Simulation of Turbulence., Cambridge University Press, Cambridge, 1999.
![]() |
[28] |
T. Dubois, F. Jauberteau and R. Temam,
Incremental unknowns, multilevel methods and the numerical simulation of turbulence, Comput. Methods Appl. Mech. Engrg., 159 (1998), 123-189.
doi: 10.1016/S0045-7825(98)80106-0. |
[29] |
S. Dumont and J.-B. Duval,
Numerical investigation of asymptotical properties of solutions to models for water waves with non local viscosity, Int. J. Numer. Anal. Model., 10 (2013), 333-349.
|
[30] |
S. Dumont and I. Manoubi,
Numerical analysis of a water wave model with a nonlocal viscous dispersive term using the diffusive approach, Math. Methods Appl. Sci., 41 (2018), 4810-4826.
|
[31] |
D. Dutykh,
Viscous-potential free-surface flows and long wave modelling, Eur. J. Mech. B Fluids, 28 (2009), 430-443.
doi: 10.1016/j.euromechflu.2008.11.003. |
[32] |
D. Dutykh and F. Dias,
Viscous potentiel free-surface flows in a fluid layer of finite depth, C. R. Math. Acad. Sci. Paris, 345 (2007), 113-118.
doi: 10.1016/j.crma.2007.06.007. |
[33] |
H. Emmerich, The Diffuse Interface Approach in Materials Science Thermodynamic, Concepts and Applications of Phase-Field Models. Lecture Notes in Physics Monographs, Springer, Heidelberg, 2003. Google Scholar |
[34] |
A. Ern, J.-L. Guermond, Theory and Practice of Finite Elements, Applied Mathematical Science, 159, Springer-Verlag, New-York, 2004.
doi: 10.1007/978-1-4757-4355-5. |
[35] |
D. J. Eyre, Unconditionallly Stable One-step Scheme for Gradient Systems, June 1998, unpublished, http://www.math.utah.edu/eyre/research/methods/stable.ps. Google Scholar |
[36] |
E. Ezzoug, O. Goubet and E. Zahrouni,
Semi-discrete weakly damped nonlinear 2-D Schrödinger equation, Differential Integral Equations, 23 (2010), 237-252.
|
[37] |
S. Faure, J. Laminie and R. Temam,
Finite volume discretization and multilevel methods in flow problems, J. Sci. Comput., 25 (2005), 231-261.
doi: 10.1007/s10915-004-4642-6. |
[38] |
, FreeFem++ Page, http://www.freefem.org Google Scholar |
[39] |
P. Garnier,
Damping to prevent the blow-up of the Korteweg-de Vries equation, Commun. Pure Appl. Anal., 16 (2017), 1455-1470.
doi: 10.3934/cpaa.2017069. |
[40] |
S. Gasparin, J. Berger, D. Dutykh and N. Mendes,
Stable explicit schemes for simulation of nonlinear moisture transfer in porous materials, J. Build. Perf. Sim., 11 (2017), 129-144.
doi: 10.1080/19401493.2017.1298669. |
[41] |
J.-M. Ghidaglia,
Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time, J. Differential Equations, 74 (1988), 369-390.
doi: 10.1016/0022-0396(88)90010-1. |
[42] |
J.-M. Ghidaglia,
A note on the strong convergence towards attractors for damped forced KdV equations, J. Differential Equations, 110 (1994), 356-359.
doi: 10.1006/jdeq.1994.1071. |
[43] |
O. Goubet,
Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations, Discrete Contin. Dynam. Systems, 6 (2000), 625-644.
doi: 10.3934/dcds.2000.6.625. |
[44] |
O. Goubet and R. M. S. Rosa,
Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line, J. Differential Equations, 185 (2002), 25-53.
doi: 10.1006/jdeq.2001.4163. |
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