June  2021, 26(6): 3335-3355. doi: 10.3934/dcdsb.2020234

Entropy-dissipating finite-difference schemes for nonlinear fourth-order parabolic equations

Institute of Analysis and Scientific Computing, Technische Universität Wien, Wiedner Hauptstraße 8–10, 1040 Wien, Austria

* Corresponding author: Ansgar Jüngel

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

Fund Project: The authors acknowledge partial support from the Austrian Science Fund (FWF), grants F65, P30000, P33010, and W1245

Structure-preserving finite-difference schemes for general nonlinear fourth-order parabolic equations on the one-dimensional torus are derived. Examples include the thin-film and the Derrida–Lebowitz–Speer–Spohn equations. The schemes conserve the mass and dissipate the entropy. The scheme associated to the logarithmic entropy also preserves the positivity. The idea of the derivation is to reformulate the equations in such a way that the chain rule is avoided. A central finite-difference discretization is then applied to the reformulation. In this way, the same dissipation rates as in the continuous case are recovered. The strategy can be extended to a multi-dimensional thin-film equation. Numerical examples in one and two space dimensions illustrate the dissipation properties.

Citation: Marcel Braukhoff, Ansgar Jüngel. Entropy-dissipating finite-difference schemes for nonlinear fourth-order parabolic equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3335-3355. doi: 10.3934/dcdsb.2020234
References:
[1]

F. Bernis, Viscous flows, fourth order nonlinear degenerate parabolic equations and singular elliptic problems, In: J.I. Díaz et al. (eds.). Free Boundary Problems: Theory and Applications. Longman Sci. Tech., Pitman Res. Notes Math. Ser., 323 (1995), 40–56.

[2]

A. L. Bertozzi and J. B. Greer, Low-curvature image simplifiers: Global regularity of smooth solutions and Laplacian limiting schemes, Commun. Pure Appl. Math., 57 (2004), 764-790.  doi: 10.1002/cpa.20019.

[3]

A. L. Bertozzi and M. Pugh, The lubrication approximation for thin viscous films: regularity and long-time behavior of weak solutions, Commun. Pure Appl. Math., 49 (1996), 85-123.  doi: 10.1002/(SICI)1097-0312(199602)49:2<85::AID-CPA1>3.0.CO;2-2.

[4]

P. M. BleherJ. L. Lebowitz and E. R. Speer, Existence and positivity of solutions of a fourth-order nonlinear PDE describing interface fluctuations, Commun. Pure Appl. Math., 47 (1994), 923-942.  doi: 10.1002/cpa.3160470702.

[5]

A.-S. BoudouP. CaputoP. Dai Pra and G. Posta, Spectral gap estimates for interacting particle systems via a Bochner-type identity, J. Funct. Anal., 232 (2006), 222-258.  doi: 10.1016/j.jfa.2005.07.012.

[6]

M. BukalE. Emmrich and A. Jüngel, Entropy-stable and entropy-dissipative approximations of a fourth-order quantum diffusion equation, Numer. Math., 127 (2014), 365-396.  doi: 10.1007/s00211-013-0588-7.

[7]

P. CaputoP. Dai Pra and G. Posta, Convex entropy decay via the Bochner–Bakry–Emery approach, Ann. Inst. H. Poincaré Prob. Stat., 45 (2009), 734-753.  doi: 10.1214/08-AIHP183.

[8]

R. Dal PassoH. Garcke and G. Grün, On a fourth order degenerate parabolic equation: global entropy estimates and qualitative behaviour of solutions, SIAM J. Math. Anal., 29 (1998), 321-342.  doi: 10.1137/S0036141096306170.

[9]

B. DerridaJ. L. LebowitzE. R. Speer and H. Spohn, Fluctuations of a stationary nonequilibrium interface, Phys. Rev. Lett., 67 (1991), 165-168.  doi: 10.1103/PhysRevLett.67.165.

[10]

B. DüringD. Matthes and J.-P. Milišić, A gradient flow scheme for nonlinear fourth order equations, Discrete Cont. Dyn. Sys. B, 14 (2010), 935-959.  doi: 10.3934/dcdsb.2010.14.935.

[11]

H. Egger, Structure preserving approximation of dissipative evolution problems, Numer. Math., 143 (2019), 85-106.  doi: 10.1007/s00211-019-01050-w.

[12]

M. Fathi and J. Maas, Entropic Ricci curvature bounds for discrete interacting systems, Ann. Appl. Prob., 26 (2016), 1774-1806.  doi: 10.1214/15-AAP1133.

[13]

D. Furihata and T. Matsuo, Discrete Variational Derivative Method, Chapman and Hall/CRC Press, Boca Raton, Florida, 2010.

[14]

P. Guidotti and K. Longo, Well-posedness for a class of fourth order diffusions for image processing, Nonlin. Diff. Eqs. Appl. NoDEA, 18 (2011), 407-425.  doi: 10.1007/s00030-011-0101-x.

[15]

X. Huo and H. Liu, A positivity-preserving and energy stable scheme for a quantum diffusion equation, Submitted for publication, 2019. arXiv: 1912.00813.

[16]

A. Jüngel and D. Matthes, The Derrida–Lebowitz–Speer–Spohn equation: Existence, non-uniqueness, and decay rates of the solutions, SIAM J. Math. Anal., 39 (2008), 1996-2015.  doi: 10.1137/060676878.

[17]

A. Jüngel and D. Matthes, An algorithmic construction of entropies in higher-order nonlinear PDEs, Nonlinearity, 19 (2006), 633-659.  doi: 10.1088/0951-7715/19/3/006.

[18]

A. Jüngel and W. Yue, Discrete Bochner inequalities via the Bochner–Bakry–Emery approach for Markov chains, Ann. Appl. Prob., 27 (2017), 2238-2269.  doi: 10.1214/16-AAP1258.

[19]

A. Jüngel and S. Schuchnigg, Entropy-dissipating semi-discrete Runge-Kutta schemes for nonlinear diffusion equations, Commun. Math. Sci., 15 (2017), 27-53.  doi: 10.4310/CMS.2017.v15.n1.a2.

[20]

A. Jüngel and J.-P. Miličić, Entropy dissipative one-leg multistep time approximations of nonlinear diffusive equations, Numer. Meth. Partial Diff. Eqs., 31 (2015), 1119-1149.  doi: 10.1002/num.21938.

[21]

S. LisiniD. Matthes and and G. Savaré, Cahn–Hilliard and thin film equations with nonlinear mobility as gradient flows in weighted-Wasserstein metrics, J. Diff. Eqs., 253 (2012), 814-850.  doi: 10.1016/j.jde.2012.04.004.

[22]

J. Maas and D. Matthes, Long-time behavior of a finite volume discretization for a fourth order diffusion equation, Nonlinearity, 29 (2016), 1992-2023.  doi: 10.1088/0951-7715/29/7/1992.

[23]

D. Matthes and H. Osberger, A convergent Lagrangian discretization for a nonlinear fourth-order equation, Found. Comput. Math., 17 (2017), 73-126.  doi: 10.1007/s10208-015-9284-6.

[24]

G. W. Wei, Generalized Perona–Malik equation for image restoration, IEEE Signal Process. Lett., 6 (1999), 165-167.  doi: 10.1109/97.769359.

[25]

L. Zhornitskaya and A. L. Bertozzi, Positivity-preserving numerical schemes for lubrication-type equations, SIAM J. Numer. Anal., 37 (2000), 523-555.  doi: 10.1137/S0036142998335698.

show all references

References:
[1]

F. Bernis, Viscous flows, fourth order nonlinear degenerate parabolic equations and singular elliptic problems, In: J.I. Díaz et al. (eds.). Free Boundary Problems: Theory and Applications. Longman Sci. Tech., Pitman Res. Notes Math. Ser., 323 (1995), 40–56.

[2]

A. L. Bertozzi and J. B. Greer, Low-curvature image simplifiers: Global regularity of smooth solutions and Laplacian limiting schemes, Commun. Pure Appl. Math., 57 (2004), 764-790.  doi: 10.1002/cpa.20019.

[3]

A. L. Bertozzi and M. Pugh, The lubrication approximation for thin viscous films: regularity and long-time behavior of weak solutions, Commun. Pure Appl. Math., 49 (1996), 85-123.  doi: 10.1002/(SICI)1097-0312(199602)49:2<85::AID-CPA1>3.0.CO;2-2.

[4]

P. M. BleherJ. L. Lebowitz and E. R. Speer, Existence and positivity of solutions of a fourth-order nonlinear PDE describing interface fluctuations, Commun. Pure Appl. Math., 47 (1994), 923-942.  doi: 10.1002/cpa.3160470702.

[5]

A.-S. BoudouP. CaputoP. Dai Pra and G. Posta, Spectral gap estimates for interacting particle systems via a Bochner-type identity, J. Funct. Anal., 232 (2006), 222-258.  doi: 10.1016/j.jfa.2005.07.012.

[6]

M. BukalE. Emmrich and A. Jüngel, Entropy-stable and entropy-dissipative approximations of a fourth-order quantum diffusion equation, Numer. Math., 127 (2014), 365-396.  doi: 10.1007/s00211-013-0588-7.

[7]

P. CaputoP. Dai Pra and G. Posta, Convex entropy decay via the Bochner–Bakry–Emery approach, Ann. Inst. H. Poincaré Prob. Stat., 45 (2009), 734-753.  doi: 10.1214/08-AIHP183.

[8]

R. Dal PassoH. Garcke and G. Grün, On a fourth order degenerate parabolic equation: global entropy estimates and qualitative behaviour of solutions, SIAM J. Math. Anal., 29 (1998), 321-342.  doi: 10.1137/S0036141096306170.

[9]

B. DerridaJ. L. LebowitzE. R. Speer and H. Spohn, Fluctuations of a stationary nonequilibrium interface, Phys. Rev. Lett., 67 (1991), 165-168.  doi: 10.1103/PhysRevLett.67.165.

[10]

B. DüringD. Matthes and J.-P. Milišić, A gradient flow scheme for nonlinear fourth order equations, Discrete Cont. Dyn. Sys. B, 14 (2010), 935-959.  doi: 10.3934/dcdsb.2010.14.935.

[11]

H. Egger, Structure preserving approximation of dissipative evolution problems, Numer. Math., 143 (2019), 85-106.  doi: 10.1007/s00211-019-01050-w.

[12]

M. Fathi and J. Maas, Entropic Ricci curvature bounds for discrete interacting systems, Ann. Appl. Prob., 26 (2016), 1774-1806.  doi: 10.1214/15-AAP1133.

[13]

D. Furihata and T. Matsuo, Discrete Variational Derivative Method, Chapman and Hall/CRC Press, Boca Raton, Florida, 2010.

[14]

P. Guidotti and K. Longo, Well-posedness for a class of fourth order diffusions for image processing, Nonlin. Diff. Eqs. Appl. NoDEA, 18 (2011), 407-425.  doi: 10.1007/s00030-011-0101-x.

[15]

X. Huo and H. Liu, A positivity-preserving and energy stable scheme for a quantum diffusion equation, Submitted for publication, 2019. arXiv: 1912.00813.

[16]

A. Jüngel and D. Matthes, The Derrida–Lebowitz–Speer–Spohn equation: Existence, non-uniqueness, and decay rates of the solutions, SIAM J. Math. Anal., 39 (2008), 1996-2015.  doi: 10.1137/060676878.

[17]

A. Jüngel and D. Matthes, An algorithmic construction of entropies in higher-order nonlinear PDEs, Nonlinearity, 19 (2006), 633-659.  doi: 10.1088/0951-7715/19/3/006.

[18]

A. Jüngel and W. Yue, Discrete Bochner inequalities via the Bochner–Bakry–Emery approach for Markov chains, Ann. Appl. Prob., 27 (2017), 2238-2269.  doi: 10.1214/16-AAP1258.

[19]

A. Jüngel and S. Schuchnigg, Entropy-dissipating semi-discrete Runge-Kutta schemes for nonlinear diffusion equations, Commun. Math. Sci., 15 (2017), 27-53.  doi: 10.4310/CMS.2017.v15.n1.a2.

[20]

A. Jüngel and J.-P. Miličić, Entropy dissipative one-leg multistep time approximations of nonlinear diffusive equations, Numer. Meth. Partial Diff. Eqs., 31 (2015), 1119-1149.  doi: 10.1002/num.21938.

[21]

S. LisiniD. Matthes and and G. Savaré, Cahn–Hilliard and thin film equations with nonlinear mobility as gradient flows in weighted-Wasserstein metrics, J. Diff. Eqs., 253 (2012), 814-850.  doi: 10.1016/j.jde.2012.04.004.

[22]

J. Maas and D. Matthes, Long-time behavior of a finite volume discretization for a fourth order diffusion equation, Nonlinearity, 29 (2016), 1992-2023.  doi: 10.1088/0951-7715/29/7/1992.

[23]

D. Matthes and H. Osberger, A convergent Lagrangian discretization for a nonlinear fourth-order equation, Found. Comput. Math., 17 (2017), 73-126.  doi: 10.1007/s10208-015-9284-6.

[24]

G. W. Wei, Generalized Perona–Malik equation for image restoration, IEEE Signal Process. Lett., 6 (1999), 165-167.  doi: 10.1109/97.769359.

[25]

L. Zhornitskaya and A. L. Bertozzi, Positivity-preserving numerical schemes for lubrication-type equations, SIAM J. Numer. Anal., 37 (2000), 523-555.  doi: 10.1137/S0036142998335698.

Figure 1.  Evolution of the DLSS equation in a semi-logarithmic scale, using the initial datum $ u^0(x) = \max\{10^{-10}, \cos(\pi x)^{16}\} $
Figure 2.  Left: Decay of the logarithmic entropy $ s_0(u(t)) $ for two different space grid sizes $ h = 1/20 $ and $ h = 1/200 $. Right: Convergence of the $ \ell^2 $ error. The dots are the values from the numerical solution, the solid line is the regression curve
Figure 3.  Left: Decay of the Shannon entropy $ s_1(u(t)) $ with $ h = 1/100 $. Right: Convergence of the $ \ell^2 $ error. The dots are the values from the numerical solution, the solid line is the regression curve
Figure 4.  Evolution of the solution to the thin-film equation at times $ t = 0 $ (densely dotted), $ t = 2\cdot 10^{-4} $ (dotted), $ t = 5\cdot 10^{-4} $ (dash-dotted), $ t = 1\cdot 10^{-3} $ (dashed), $ t = 2\cdot 10^{-3} $ (densely dashed), and $ t = 5\cdot 10^{-3} $ (solid) and grid sizes $ h = 1/10 $ (left), $ h = 1/200 $ (right)
Figure 5.  Decay of the logarithmic entropy $ S_0(u(t)) $ for various space grid sizes
Figure 6.  Evolution of the solution to the two-dimensional thin-film equation with $ \beta = 2 $, $ t = 0 $ (top left), $ t = 3\cdot 10^{-9} $ (top right), $ t = 10^{-8} $ (bottom left), $ t = 10^{-6} $ (bottom right)
Figure 7.  Decay of the logarithmic entropy $ S_0(u(t)) $ for various space grid sizes
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