Theoretical and computational analysis of a nonlinear Schrödinger problem with moving boundary

Coupled parabolic-hyperbolic system often appears in the studies of thermoelasticity, magnetoelasticity, biological problems and radiation hydrodynamics with high temperature. In this paper, we investigate a problem of thermal diffusion in elastic body with moving boundary. Three numerical methods, two uncoupled and one coupled, all with quadratic convergence order in time and space, are presented and compared in relation to the execution time. Tables and figures of the approximate solution are shown to verify the efficiency and feasibility of the proposed method. In addition, we show that the numerical results are consistent with the theoretical results. The approximate numerical solutions are calculated using the finite element method in spatial variable and finite difference method in time. To compare different numerical schemes, we show the convergence rates through numerical simulations with calculated error and the execution time. KEYWORDScoupled and uncoupled method, finite elements method, moving boundary, numerical simulation, thermal diffusion INTRODUCTIONThe thermal diffusion in elastic body is governed by a coupled system of hyperbolic-parabolic equations. In [1], the theory was developed using the coupled thermoelastic model. Uniqueness and reciprocity theorems for the generalized problem of thermal diffusion in elastic body in isotropic media, was proved in [2], under restrictive assumptions on the elastic coefficients. In these works, one

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“…Let also the moving boundary defined by the function (other nonlinear boundary types can be found, for example in [17,18]):…”

Section: Examplementioning

confidence: 99%

Numerical methods for a problem of thermal diffusion in elastic body with moving boundary

Madureira

Rincon

Teixeira

2021

Numerical Methods Partial

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“…These results can help numerical and computational advances in other types of equilibrium problems with controls acting on the moving limit, such as Stackelberg-Pareto, Pareto, Nash, among others. It can also be extended into similar analyses for other types of hierarchical control problems, such as Heat equation, Stokes, Navier-Stokes, Schrödinger (in [14] some results are presented), among others.…”

Section: Some Additional Comments and Conclusionmentioning

confidence: 99%

On the Computation of Hierarchical Control results for One-Dimensional Transmission Line

de Carvalho,

Neto

2021

Preprint

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In this paper, motivated by a physics problem, we investigate some numerical and computational aspects for the problem of hierarchical controllability in a one-dimensional wave equation in domains with a moving boundary. Some controls act in part of the boundary and define a strategy of equilibrium between them, considering a leader control and a follower. Thus, we introduced the concept of hierarchical control to solve the problem and mapped the Stackelberg Strategy between these controls. A total discretization of the problem is presented for a numerical evaluation in spaces of finite dimension, an algorithm for evaluation of the problem is presented as the combination of finite element method (FEM) and finite difference method (FDM). The algorithm efficiency and computational results are illustrated for some experiments using the software FreeFem ++ .

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Theoretical and numerical study of a Burgers viscous equation type with moving boundary

Pereira,

Carmo,

Rincon

et al. 2024

Math Methods in App Sciences

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In this article, we investigate the existence, uniqueness, and numerical aspects of a one‐ and two‐dimensional nonlinear viscous type Burgers problem defined in a noncylindrical domain. In order to obtain the existence and uniqueness of the solution, the problem with a moving ends is transformed into an equivalent problem in a cylindrical through a diffeomorphism between the domains. The numerical simulation for the one‐ and two‐dimensional cases is performed using Lagrange with degrees 1–3 and cubic Hermite polynomials as base functions for applying the linearized Crank–Nicolson–Galerkin method to obtain an approximate numerical solution. Graphs prove the efficiency of the numerical method along with the order of numerical convergence consistent with the degree of the base polynomial.

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Theoretical and computational analysis of a nonlinear Schrödinger problem with moving boundary

Cited by 6 publications

References 13 publications

Numerical methods for a problem of thermal diffusion in elastic body with moving boundary

Numerical methods for a problem of thermal diffusion in elastic body with moving boundary

On the Computation of Hierarchical Control results for One-Dimensional Transmission Line

Theoretical and numerical study of a Burgers viscous equation type with moving boundary

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