# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021153
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## 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

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:
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##### References:
 [1] M. Bonafini, M. 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. Chen, S. 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. Yoshiuchi, S. Omata, K. Svadlenka and K. Ohara, Numerical solution of film vibration with obstacle, Adv. Math. Sci. Appl., 16 (2006), 33-43.   Google Scholar
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 $\Delta x = h$
. 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 $\varepsilon >0$
Evolution of the energy of the numerical solution for both methods
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
Comparison of the Crank-Nicolson scheme with the original discrete Morse flow for a 2-dimensional problem
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
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
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