• Previous Article
    On the generalized Burgers-Huxley equation: Existence, uniqueness, regularity, global attractors and numerical studies
  • DCDS-B Home
  • This Issue
  • Next Article
    A dynamical theory for singular stochastic delay differential equations Ⅱ: nonlinear equations and invariant manifolds
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  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 & Continuous Dynamical Systems - B, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

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
[1]

Murat Uzunca, Ayşe Sarıaydın-Filibelioǧlu. Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 269-281. doi: 10.3934/naco.2020025

[2]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068

[3]

Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213

[4]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437

[5]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109

[6]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25

[7]

Jumpei Inoue, Kousuke Kuto. On the unboundedness of the ratio of species and resources for the diffusive logistic equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2441-2450. doi: 10.3934/dcdsb.2020186

[8]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309

[9]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258

[10]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1649-1672. doi: 10.3934/dcdss.2020448

[11]

Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1779-1799. doi: 10.3934/dcdss.2020454

[12]

V. V. Zhikov, S. E. Pastukhova. Korn inequalities on thin periodic structures. Networks & Heterogeneous Media, 2009, 4 (1) : 153-175. doi: 10.3934/nhm.2009.4.153

[13]

Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175

[14]

Ademir Fernando Pazoto, Lionel Rosier. Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1511-1535. doi: 10.3934/dcdsb.2010.14.1511

[15]

Abdulrazzaq T. Abed, Azzam S. Y. Aladool. Applying particle swarm optimization based on Padé approximant to solve ordinary differential equation. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021008

[16]

Vo Anh Khoa, Thi Kim Thoa Thieu, Ekeoma Rowland Ijioma. On a pore-scale stationary diffusion equation: Scaling effects and correctors for the homogenization limit. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2451-2477. doi: 10.3934/dcdsb.2020190

[17]

Caifang Wang, Tie Zhou. The order of convergence for Landweber Scheme with $\alpha,\beta$-rule. Inverse Problems & Imaging, 2012, 6 (1) : 133-146. doi: 10.3934/ipi.2012.6.133

[18]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825

[19]

Elena Bonetti, Pierluigi Colli, Gianni Gilardi. Singular limit of an integrodifferential system related to the entropy balance. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1935-1953. doi: 10.3934/dcdsb.2014.19.1935

[20]

Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $ \Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (49)
  • HTML views (220)
  • Cited by (0)

Other articles
by authors

[Back to Top]