Abstract
In this article, we give analysis for a structure-preserving finite difference scheme to the Cahn–Hilliard system coupled with elasticity in one space dimension. In the previous article [K. Shimura and S. Yoshikawa, Error estimate for structure-preserving finite difference schemes of the one-dimensional Cahn–Hilliard system coupled with viscoelasticity, Regularity and Asymptotic Analysis for Critical Cases of Partial Differential Equations, RIMS Kôkyûroku Bessatsu B82, Research Institute for Mathematical Sciences (RIMS), Kyoto 2020, 159–175], we studied the system coupled with viscoelasticity, where we proposed a conservative numerical scheme for the system which inherits the total energy conservation and momentum conservation laws, and showed the error estimate. However, the error estimate can not be applied to the system without viscosity, due to the fact that the proof relies on the viscous term. Here, we show the error estimate for the system without viscosity by proposing a new structure-preserving finite difference scheme for the system. In addition, we also give the proof of existence of solution for the scheme.
Funding source: Japan Society for the Promotion of Science
Award Identifier / Grant number: JP16K05234
Award Identifier / Grant number: JP20K03687
Funding source: Sumitomo Foundation
Award Identifier / Grant number: 180823
Funding statement: This work was partially supported by JSPS KAKENHI Grant Nos. JP16K05234, JP20K03687 and Sumitomo Foundation Grant No. 180823.
Acknowledgements
We express our deep gratitude to the anonymous referees for their helpful advices.
References
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© 2022 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Estimates for a beam-like partial differential operator and applications
- Stabilization of polynomial systems in ℝ3 via homogeneous feedback
- Analyzing the existence of solution of a fractional order integral equation: A fixed point approach
- Existence of solution to a nonlocal biharmonic problem with dependence on gradient and Laplacian
- Global existence and exponential decay for a viscoelastic equation with not necessarily decreasing kernel
- Solving fractal differential equations via fractal Laplace transforms
- Minimum energy control of degenerate Cauchy problem with skew-Hermitian pencil
- Splines in vibration analysis of non-homogeneous circular plates of quadratic thickness
- The weak eigenfunctions of boundary-value problem with symmetric discontinuities
- Some subclasses of analytic functions involving certain integral operator
- Relation theoretic contractions and their applications in b-metric like spaces
- A new conservative finite difference scheme for 1D Cahn–Hilliard equation coupled with elasticity
- An improved proximal method with quasi-distance for nonconvex multiobjective optimization problem
Artikel in diesem Heft
- Frontmatter
- Estimates for a beam-like partial differential operator and applications
- Stabilization of polynomial systems in ℝ3 via homogeneous feedback
- Analyzing the existence of solution of a fractional order integral equation: A fixed point approach
- Existence of solution to a nonlocal biharmonic problem with dependence on gradient and Laplacian
- Global existence and exponential decay for a viscoelastic equation with not necessarily decreasing kernel
- Solving fractal differential equations via fractal Laplace transforms
- Minimum energy control of degenerate Cauchy problem with skew-Hermitian pencil
- Splines in vibration analysis of non-homogeneous circular plates of quadratic thickness
- The weak eigenfunctions of boundary-value problem with symmetric discontinuities
- Some subclasses of analytic functions involving certain integral operator
- Relation theoretic contractions and their applications in b-metric like spaces
- A new conservative finite difference scheme for 1D Cahn–Hilliard equation coupled with elasticity
- An improved proximal method with quasi-distance for nonconvex multiobjective optimization problem