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Journal of Applied Mathematics and Computation

ISSN Print: 2576-0645 Downloads: 153774 Total View: 1839433
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Article Open Access http://dx.doi.org/10.26855/jamc.2022.12.005

Estimate of an Error for a Finite Difference a Phase Field Model for the Euler-Crank-Nicolson Methods

Chrysovalantis A. Sfyrakis1,*, George E. Chatzarakis2, Spyros L. Panetsos2

1Department of Mechanical Engineering Educators, School of Pedagogical & Technological Education (ASPETE), Marousi 15122, Athens, Greece. 

2Department of Electrical and Electronic Engineering Educators, School of Pedagogical & Technological Education (ASPETE), Marousi 15122, Athens, Greece.

*Corresponding author: Chrysovalantis A. Sfyrakis

Published: December 1,2022

Abstract

In order to understand phase transition processes like solidification, phase field models are frequently used. The energy (heat) equation for temperature is coupled with another nonlinear parabolic p.d.e. that includes a second unknown, the phase, which takes characteristic values, such as zero in the solid phase and one in the liquid phase. We consider the parabolic system of p.d.e’ s 

q(ϕ)ϕt=∇⋅(A(ϕ)∇ϕ)+f(ϕ,u),

ut=Δu+[p(ϕ)]t,

which may be considered as a simplified phase field model. Here, ϕ=ϕ(x,y,t) is the phase indicator function, u=u(x,y,t) is the temperature, q,p, and f are given scalar functions, and A is a 2×2 diagonal matrix of given functions of ϕ. This system is posed for t≥0 on a rectangle in the x,y plane with appropriate boundary and initial conditions. We solve the system using a finite difference method that uses for first equation recursive Euler and Crank-Nicolson method the other equation. We prove a convergence result for the method and show results of numerical experiments verifying its order of accuracy.

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How to cite this paper

Estimate of an Error for a Finite Difference a Phase Field Model for the Euler-Crank-Nicolson Methods

How to cite this paper:  Chrysovalantis A. Sfyrakis, George E. Chatzarakis, Spyros L. Panetsos. (2022) Estimate of an Error for a Finite Difference a Phase Field Model for the Euler-Crank-Nicolson Methods. Journal of Applied Mathematics and Computation6(4), 431-449.

DOI: http://dx.doi.org/10.26855/jamc.2022.12.005