Numerical study of one-dimensional Stefan problem with periodic boundary conditions

Abstract

A finite difference approach to a one-dimensional Stefan problem with periodic boundary conditions is studied. The evolution of the moving boundary and the temperature field are simulated numerically, and the effects of the Stefan number and the periodical boundary condition on the temperature distribution and the evolution of the moving boundary are analyzed.interface each time, namely the constant space grid size. Thus R = N∆r is the position of the moving interface at t = t N , with t 0 = 0 and N = 0, 1, … Using a backward difference scheme for the time derivative and a central difference scheme for the space derivative, eq. (1) in discrete form can be expressed:

Two numerical methods for heat conduction problems with phase change

“…We establish a finite difference scheme to solve the system (1)-(5) by considering the needed time corresponding to the forward-moving constant distance of the moving interface each time [8]. Let Dx be the forward-moving distance of the phase change interface each time, namely the constant space step size.…”

“…Caldwell and Kwan [7] presented a brief review of five key numerical methods for 1-D Stefan problems for simple geometries including plane, cylindrical and spherical, and obtained the numerical results of spherical and cylindrical phase change problem with fixed boundary condition from four methods including the enthalpy method, boundary immobilization method, perturbation method and heat balance integral method. Qu et al [8] established a finite difference method to solve the Stefan problem with periodic boundary condition and analyzed the effects of the oscillating surface temperature on the motion of the moving interface and the temperature distribution.…”

On a multiple time-scales perturbation analysis of a Stefan problem with a time-dependent Dirichlet boundary condition

“…However, exact analytical solutions are only available for some very specific cases. Furthermore, most Stefan problems are solved approximately by means of numerical methods (see for instance [6][7][8][13][14][15][16][17][18]). Another alternative to approximate the solution is by using a straightforward naive perturbation expansion (see for instance [6,7,[19][20][21][22][23][24][25]).…”

On a multiple timescales perturbation approach for a stefan problem with a time-dependent heat flux at the boundary

“…Most Stefan problems are solved approximately by using numerical methods (see for instance [4,10,12,13,16,18,19,24,25]). A smaller part of the Stefan prob-lems is approximately solved by using straightforward perturbation expansion (see for instance [3,6,9,12,13,15,17,26,28]).…”