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A new conservative finite difference scheme for 1D Cahn-Hilliard equation coupled with elasticity

Shimura Kazuki, Yoshikawa Shuji

Journal of Applied Analysis

2022

Vol. 28, nr 2

311--332

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, in: 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.

On an application of an interval backward ﬁnite diﬀerence method for solving the one-dimensional heat conduction problem

Jankowska M., Marciniak A., Hoﬀmann T.

Control and Cybernetics

2015

Vol. 44, no. 4

463--480

The paper concerns the interval method for solving the one-dimensional heat conduction problem. It is based on the conventional backward ﬁnite diﬀerence scheme with the appropriate local truncation error terms that are also taken into account. For the theoretical formulation of the interval approach we can show that the exact solution is included in the interval one. In practice, there are problems, for which we cannot determine the endpoints of the error term intervals exactly. Nevertheless, if we use the appropriate approximation, related to the endpoints considered, then the numerical experiments conﬁrm that the interval solution includes the exact one.