Damping, stabilization, and numerical filtering for the modeling and the simulation of time dependent PDEs

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August 2021, 14(8): 2693-2728. doi: 10.3934/dcdss.2021002

Damping, stabilization, and numerical filtering for the modeling and the simulation of time dependent PDEs

Jean-Paul Chehab ^,

Laboratoire LAMFA (UMR CNRS 7352), Université de Picardie Jules Verne, 33 rue Saint Leu, 80039 Amiens Cédex, France

^* Corresponding author: Jean-Paul Chehab

To the memory of Ezzeddine Zahrouni (1963-2018)

Received June 2020 Revised December 2020 Published August 2021 Early access January 2021

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.

Keywords: Cahn-Hilliard equation, Korteweig-de Vries equation, Kuramoto-Sivashinski equation, finite differences compact Schemes, finite differences, multigrid methods, stability, numerical filters, damping modelling.

Mathematics Subject Classification: Primary:35B40, 35K55, 65M06, 65M55, 65M60.

Citation: Jean-Paul Chehab. Damping, stabilization, and numerical filtering for the modeling and the simulation of time dependent PDEs. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 2693-2728. doi: 10.3934/dcdss.2021002

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).
[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.
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[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.
[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.
[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.
[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.
[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.
[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.
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[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.
[45]	O. Goubet and E. Zahrouni, On a time discretization of a weakly damped forced nonlinear Schrödinger equation, Commun. Pure Appl. Anal., 7 (2008), 1429-1442. doi: 10.3934/cpaa.2008.7.1429.
[46]	J. M. Hyman and B. Nikolaenko, The Kuramoto-Sivashinky equation: A bridge between PDE's and dynamical systems, Physica D, 18 (1986), 113-126. doi: 10.1016/0167-2789(86)90166-1.
[47]	M. S. Jolly, R. Rosa and R. Temam, Accurate computations on inertial manifolds, SIAM J. Sci. Comput., 22 (2000), 2216-2238. doi: 10.1137/S1064827599351738.
[48]	C. Klein and R. Peter, Numerical study of blow-up and dispersive shocks in solutions to generalized Korteweg-de Vries equations, Phys. D, 304/305 (2015), 52-78. doi: 10.1016/j.physd.2015.04.003.
[49]	C. Laurent, L. Rosier and B.-Y. Zhang, Control stabilization of the Korterweg-de Vries equation in a periodic domain, Comm. Partial Differential Equations, 35 (2010), 707-744. doi: 10.1080/03605300903585336.
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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).
[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.
[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.
[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.
[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.
[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.
[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
[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.
[45]	O. Goubet and E. Zahrouni, On a time discretization of a weakly damped forced nonlinear Schrödinger equation, Commun. Pure Appl. Anal., 7 (2008), 1429-1442. doi: 10.3934/cpaa.2008.7.1429.
[46]	J. M. Hyman and B. Nikolaenko, The Kuramoto-Sivashinky equation: A bridge between PDE's and dynamical systems, Physica D, 18 (1986), 113-126. doi: 10.1016/0167-2789(86)90166-1.
[47]	M. S. Jolly, R. Rosa and R. Temam, Accurate computations on inertial manifolds, SIAM J. Sci. Comput., 22 (2000), 2216-2238. doi: 10.1137/S1064827599351738.
[48]	C. Klein and R. Peter, Numerical study of blow-up and dispersive shocks in solutions to generalized Korteweg-de Vries equations, Phys. D, 304/305 (2015), 52-78. doi: 10.1016/j.physd.2015.04.003.
[49]	C. Laurent, L. Rosier and B.-Y. Zhang, Control stabilization of the Korterweg-de Vries equation in a periodic domain, Comm. Partial Differential Equations, 35 (2010), 707-744. doi: 10.1080/03605300903585336.
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[51]	C. Lemaréchal, Une méthode de résolution de certains systèmes non linéaires bien posés, C. R. Acad. Sci. Paris, sér. A, 272 (1971), 605-607.
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Figure 1. Different filters

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Figure 2. Different basis of orthogonal polynomials: Chebyshev polynomials (top left), Legendre polynomials (top right) and Fourier polynomials (bottom)

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Figure 3. Hierarchy of the triangulation of $ \Omega = ]0,1[^2 $: solid line (coarse triangulation), dashed line (complementary triangulation)

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Figure 4. 3D output and iso-values for $ u(x,y) = \cos \bigl(5(1-x^2-y^2)\bigr) $ on the unit disk. The function $ u $ with $ \mathbb{P}_2 $ elements (top) and the $ z_h $ components (bottom)

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Figure 5. Function $ u(x,y) = \cos \bigl(5(1-x^2-y^2)\bigr) $ on the unit disk. Eigenfunction (Fourier) coefficients on the fine mesh for the original function $ u_h $ (red line) and the associated correction $ z_h $ (blue line)

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Figure 6. Decomposition of the signal $ u(x) = \sin(2\pi x)+\sin(6\pi x)+\sin(12\pi x)+0.1\sin(20\pi x)+0.1\sin(30\pi x)+0.1\sin(120\pi x) $, $ N = 100, m = 8 $. Original signal (top), high frequencies fluctuent part (bottom left) and low frequencies mean part (bottom right)

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Figure 7. Dirichlet BC (Top Left), Neumannn BC (Top Right), Periodic BC (Bottom). The original values are given at grid points $ \times $, interpolated values are computed at grid points $ o $. The boundary points are marked by a dot, in the Dirichlet case

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Figure 8. Low-pass filters: (left) - Exponential of low pass filters = high-pass filter (right)

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Figure 9. Coarse mesh (left) and fine mesh (right)

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Figure 10. Initial solution (left) and Final Solution (right) - \hskip 1.cm $ \mathbb{P}_2 $ Elements. $ \epsilon=0.08 $, $ \Delta t =1.e-4, \tau=4\epsilon^2, T=0.012 $

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Figure 11. Energy (left) and mass (right) vs time - \hskip 2.cm $ \mathbb{P}_2 $ Elements. $ \epsilon=0.08 $, $ \Delta t =1.e-4, \tau=4\epsilon^2, \ T=0.012 $

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Figure 12. $ \|u\|_{L^2} $ vs time with $ u_0 = \cos(2\pi x/L)+\cos(6\pi x/L)+\cos(12\pi x/L)+\cos(20\pi x/L) $, $ (\tau_0,\tau_1) = (10,100) $ (left) $ (\tau_0,\tau_1) = (100,10) $ (right), $ \Delta t = 1.e-1 $, $ T = 10 $, $ L = 100 $

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Figure 13. KdV $ (\tau_0,\tau_1) = (0,0) $, $ \mathbb{P}_1 $ Elements. $ \Delta t = 1.e-2 $, $ T = 40 $. Mass $ \int_0^Ludx $ (left) and $ L^2 $-norm $ |u|_{L^2} $ (right) vs time

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Figure 14. KdV $ (\tau_0,\tau_1) = (0,100) $ Low pass damping, $ \mathbb{P}_1 $ Elements. $ \Delta t = 1.e-2 $, $ T = 40 $. Mass $ \int_0^Ludx $ (left) and $ L^2 $-norm $ |u|_{L^2} $ (right) vs time

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Figure 15. KdV $ (\tau_0,\tau_1) = (0,100) $ Low pass damping, $ \mathbb{P}_1 $ Elements. $ \Delta t = 1.e-2 $, $ T = 4 $. Mass $ \int_0^Ludx $ (left) and $ L^2 $-norm $ |u|_{L^2} $ (right) vs time

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Figure 16. Heat Equation - $ L^{\infty} $-norm of the error vs time - $ n = 100, m = 4, \Delta t = 9.95 \, 10^{-5}, \tau = 1.6,10^{4} $

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Figure 17. KSE Low and high frequency components of the solution at final time $ T=140 $ (left), time evolution of the mean value (right) - $ n=128, \ m=8 $, $ \Delta t=0.01 $, $ L=10 $ (line 1), $ L=20 $ (line 2)

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Figure 18. KSE Low and high frequency components of the solution at final time $ T=140 $ (left), time evolution of the mean value (right) - $ \Delta t=0.01 $. Line 1: $ L=50 $, $ n=128, \ m=8 $; line 2: $ L=100 $, $ n=256, \, m=10 $

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