doi: 10.3934/dcdsb.2021153
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

A Crank-Nicolson type minimization scheme for a hyperbolic free boundary problem

1. 

National Institute of Technology, Gifu College, Motosu, Gifu, 501-0495, Japan

2. 

Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, Nakanoku, Tokyo, 164-8525, Japan

3. 

Graduate School of Natural Science and Technology, Kanazawa University, Kanazawa, Ishikawa, 920-1192, Japan

4. 

Institute of Science and Engineering, Kanazawa University, Kanazawa, Ishikawa, 920-1192, Japan

5. 

Graduate School of Science, Kyoto University, Sakyoku, Kyoto, 606-8502, Japan

* Corresponding author: Karel Svadlenka

Received  April 2020 Revised  March 2021 Early access June 2021

We consider a hyperbolic free boundary problem by means of minimizing time discretized functionals of Crank-Nicolson type. The feature of this functional is that it enjoys energy conservation in the absence of free boundaries, which is an essential property for numerical calculations. The existence and regularity of minimizers is shown and an energy estimate is derived. These results are then used to show the existence of a weak solution to the free boundary problem in the 1-dimensional setting.

Citation: Yoshiho Akagawa, Elliott Ginder, Syota Koide, Seiro Omata, Karel Svadlenka. A Crank-Nicolson type minimization scheme for a hyperbolic free boundary problem. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021153
References:
[1]

M. BonafiniM. Novaga and G. Orlandi, A variational scheme for hyperbolic obstacle problems, Nonlin. Anal., 188 (2019), 389-404.  doi: 10.1016/j.na.2019.06.008.  Google Scholar

[2]

M. Bonafini, V.P.C. Le, M. Novaga and G. Orlandi, On the obstacle problem for fractional semilinear wave equations, Nonlinear Anal., 210 (2021), 112368. doi: 10.1016/j.na.2021.112368.  Google Scholar

[3]

X. ChenS. Cui and A. Friedman, A hyperbolic free boundary problem modeling tumor growth: Asymptotic behavior, Trans. Am. Math. Soc., 357 (2005), 4771-4804.  doi: 10.1090/S0002-9947-05-03784-0.  Google Scholar

[4] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton University Press, 1983.   Google Scholar
[5]

E. Ginder and K. Svadlenka, A variational approach to a constrained hyperbolic free boundary problem, Nonlinear Anal., 71 (2009), e1527–e1537. doi: 10.1016/j.na.2009.01.228.  Google Scholar

[6]

T. Iguchi and D. Lannes, Hyperbolic free boundary problems and applications to wave-structure interactions, Indiana Univ. Math. J., 70 (2021), 353–464, arXiv: 1806.07704.  Google Scholar

[7]

K. Kikuchi, Constructing a solution in time semidiscretization method to an equation of vibrating string with an obstacle, Nonlinear Anal., 71 (2009), 1227-1232.   Google Scholar

[8]

K. Kikuchi and S. Omata, A free boundary problem for a one dimensional hyperbolic equation, Adv. Math. Sci. Appl., 9 (1999), 775-786.   Google Scholar

[9] O. Ladyzhenskaya and N. Uraltseva, Linear and Quasilinear Elliptic Equations, 1$^st$ edition, Academic Press, 1968.   Google Scholar
[10]

S. Omata, A numerical treatment of film motion with free boundary, Adv. Math. Sci. Appl., 14 (2004), 129-137.   Google Scholar

[11]

S. Omata, A hyperbolic obstacle problem with an adhesion force, in Mathematics for Nonlinear Phenomena-Analysis and Computation (eds Y. Maekawa and S. Jimbo), Springer Proceedings in Mathematics and Statistics, 215 (2017), 261–269. doi: 10.1007/978-3-319-66764-5_12.  Google Scholar

[12]

K. Svadlenka and S. Omata, Mathematical modeling of surface vibration with volume constraint and its analysis, Nonlinear Anal., 69 (2008), 3202-3212.  doi: 10.1016/j.na.2007.09.013.  Google Scholar

[13]

H. YoshiuchiS. OmataK. Svadlenka and K. Ohara, Numerical solution of film vibration with obstacle, Adv. Math. Sci. Appl., 16 (2006), 33-43.   Google Scholar

show all references

References:
[1]

M. BonafiniM. Novaga and G. Orlandi, A variational scheme for hyperbolic obstacle problems, Nonlin. Anal., 188 (2019), 389-404.  doi: 10.1016/j.na.2019.06.008.  Google Scholar

[2]

M. Bonafini, V.P.C. Le, M. Novaga and G. Orlandi, On the obstacle problem for fractional semilinear wave equations, Nonlinear Anal., 210 (2021), 112368. doi: 10.1016/j.na.2021.112368.  Google Scholar

[3]

X. ChenS. Cui and A. Friedman, A hyperbolic free boundary problem modeling tumor growth: Asymptotic behavior, Trans. Am. Math. Soc., 357 (2005), 4771-4804.  doi: 10.1090/S0002-9947-05-03784-0.  Google Scholar

[4] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton University Press, 1983.   Google Scholar
[5]

E. Ginder and K. Svadlenka, A variational approach to a constrained hyperbolic free boundary problem, Nonlinear Anal., 71 (2009), e1527–e1537. doi: 10.1016/j.na.2009.01.228.  Google Scholar

[6]

T. Iguchi and D. Lannes, Hyperbolic free boundary problems and applications to wave-structure interactions, Indiana Univ. Math. J., 70 (2021), 353–464, arXiv: 1806.07704.  Google Scholar

[7]

K. Kikuchi, Constructing a solution in time semidiscretization method to an equation of vibrating string with an obstacle, Nonlinear Anal., 71 (2009), 1227-1232.   Google Scholar

[8]

K. Kikuchi and S. Omata, A free boundary problem for a one dimensional hyperbolic equation, Adv. Math. Sci. Appl., 9 (1999), 775-786.   Google Scholar

[9] O. Ladyzhenskaya and N. Uraltseva, Linear and Quasilinear Elliptic Equations, 1$^st$ edition, Academic Press, 1968.   Google Scholar
[10]

S. Omata, A numerical treatment of film motion with free boundary, Adv. Math. Sci. Appl., 14 (2004), 129-137.   Google Scholar

[11]

S. Omata, A hyperbolic obstacle problem with an adhesion force, in Mathematics for Nonlinear Phenomena-Analysis and Computation (eds Y. Maekawa and S. Jimbo), Springer Proceedings in Mathematics and Statistics, 215 (2017), 261–269. doi: 10.1007/978-3-319-66764-5_12.  Google Scholar

[12]

K. Svadlenka and S. Omata, Mathematical modeling of surface vibration with volume constraint and its analysis, Nonlinear Anal., 69 (2008), 3202-3212.  doi: 10.1016/j.na.2007.09.013.  Google Scholar

[13]

H. YoshiuchiS. OmataK. Svadlenka and K. Ohara, Numerical solution of film vibration with obstacle, Adv. Math. Sci. Appl., 16 (2006), 33-43.   Google Scholar

Figure 1.  Numerical solution at four distinct times for the Crank-Nicolson method (blue) and the original discrete Morse flow method (red). The time step size is $ h = 1.0 \times 10^{-4} $ and the spacial mesh size is $ \Delta x = h $
Figure 1. The curves are obtained by plotting the boundary of the set $ \{(t,x);u(x,t) < \varepsilon\} $ for a small $ \varepsilon >0 $">Figure 2.  Free boundary corresponding to the motion in Figure 1. The curves are obtained by plotting the boundary of the set $ \{(t,x);u(x,t) < \varepsilon\} $ for a small $ \varepsilon >0 $
Figure 3.  Evolution of the energy of the numerical solution for both methods
Figure 4.  Comparison of energy decay tendency for both methods using the initial data $ u_0 = \sin(2n\pi x) $ and $ v_0 \equiv 0 $. Here, $ \Delta x = h $ is used
Figure 5.  Comparison of the Crank-Nicolson scheme with the original discrete Morse flow for a 2-dimensional problem
Figure 6.  Crank-Nicolson type minimizing movement approximation of droplet motion, with the free boundary illustrated as the black curves. Time is designated by the integer values within the figure, so that the initial condition corresponds to number 1 and all graphs are plotted at equal time intervals, except for the last one showing the stationary state reached after sufficiently long time
Table 1.  Main features of the two methods compared in this section
C-N DMF
energy conserved decays
free boundary condition holds holds
high harmonic wave preserved decays
including constraints possible possible
phase shift occurs occurs
C-N DMF
energy conserved decays
free boundary condition holds holds
high harmonic wave preserved decays
including constraints possible possible
phase shift occurs occurs
[1]

Yingwen Guo, Yinnian He. Fully discrete finite element method based on second-order Crank-Nicolson/Adams-Bashforth scheme for the equations of motion of Oldroyd fluids of order one. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2583-2609. doi: 10.3934/dcdsb.2015.20.2583

[2]

Sondre Tesdal Galtung. A convergent Crank-Nicolson Galerkin scheme for the Benjamin-Ono equation. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1243-1268. doi: 10.3934/dcds.2018051

[3]

Dongho Kim, Eun-Jae Park. Adaptive Crank-Nicolson methods with dynamic finite-element spaces for parabolic problems. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 873-886. doi: 10.3934/dcdsb.2008.10.873

[4]

Constantine M. Dafermos. A variational approach to the Riemann problem for hyperbolic conservation laws. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 185-195. doi: 10.3934/dcds.2009.23.185

[5]

Julius Fergy T. Rabago, Hideyuki Azegami. A new energy-gap cost functional approach for the exterior Bernoulli free boundary problem. Evolution Equations & Control Theory, 2019, 8 (4) : 785-824. doi: 10.3934/eect.2019038

[6]

Jesús Ildefonso Díaz. On the free boundary for quenching type parabolic problems via local energy methods. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1799-1814. doi: 10.3934/cpaa.2014.13.1799

[7]

Robert I. McLachlan, G. R. W. Quispel. Discrete gradient methods have an energy conservation law. Discrete & Continuous Dynamical Systems, 2014, 34 (3) : 1099-1104. doi: 10.3934/dcds.2014.34.1099

[8]

Hua Chen, Wenbin Lv, Shaohua Wu. A free boundary problem for a class of parabolic type chemotaxis model. Kinetic & Related Models, 2015, 8 (4) : 667-684. doi: 10.3934/krm.2015.8.667

[9]

Hua Chen, Wenbin Lv, Shaohua Wu. A free boundary problem for a class of parabolic-elliptic type chemotaxis model. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2577-2592. doi: 10.3934/cpaa.2018122

[10]

Fuke Wu, Xuerong Mao, Peter E. Kloeden. Discrete Razumikhin-type technique and stability of the Euler--Maruyama method to stochastic functional differential equations. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 885-903. doi: 10.3934/dcds.2013.33.885

[11]

Xavier Litrico, Vincent Fromion, Gérard Scorletti. Robust feedforward boundary control of hyperbolic conservation laws. Networks & Heterogeneous Media, 2007, 2 (4) : 717-731. doi: 10.3934/nhm.2007.2.717

[12]

Herbert Gajewski, Jens A. Griepentrog. A descent method for the free energy of multicomponent systems. Discrete & Continuous Dynamical Systems, 2006, 15 (2) : 505-528. doi: 10.3934/dcds.2006.15.505

[13]

Bastian Gebauer, Nuutti Hyvönen. Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem. Inverse Problems & Imaging, 2008, 2 (3) : 355-372. doi: 10.3934/ipi.2008.2.355

[14]

Stefano Bianchini. On the shift differentiability of the flow generated by a hyperbolic system of conservation laws. Discrete & Continuous Dynamical Systems, 2000, 6 (2) : 329-350. doi: 10.3934/dcds.2000.6.329

[15]

Zhi-Qiang Shao. Lifespan of classical discontinuous solutions to the generalized nonlinear initial-boundary Riemann problem for hyperbolic conservation laws with small BV data: shocks and contact discontinuities. Communications on Pure & Applied Analysis, 2015, 14 (3) : 759-792. doi: 10.3934/cpaa.2015.14.759

[16]

Tong Yang, Fahuai Yi. Global existence and uniqueness for a hyperbolic system with free boundary. Discrete & Continuous Dynamical Systems, 2001, 7 (4) : 763-780. doi: 10.3934/dcds.2001.7.763

[17]

Yang Zhang. A free boundary problem of the cancer invasion. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021092

[18]

Toyohiko Aiki. A free boundary problem for an elastic material. Conference Publications, 2007, 2007 (Special) : 10-17. doi: 10.3934/proc.2007.2007.10

[19]

Mapundi K. Banda, Michael Herty. Numerical discretization of stabilization problems with boundary controls for systems of hyperbolic conservation laws. Mathematical Control & Related Fields, 2013, 3 (2) : 121-142. doi: 10.3934/mcrf.2013.3.121

[20]

Boris Andreianov, Mohamed Karimou Gazibo. Explicit formulation for the Dirichlet problem for parabolic-hyperbolic conservation laws. Networks & Heterogeneous Media, 2016, 11 (2) : 203-222. doi: 10.3934/nhm.2016.11.203

2020 Impact Factor: 1.327

Article outline

Figures and Tables

[Back to Top]