Regular Domain Iterative Collocation Method in Space-Time Region for Moving Boundary Problems

doi:10.19596/j.cnki.1001-246x.8177

Abstract

Abstract: The governing equation of moving boundary problem is heat conductive equation. Its definite solution domain varies with time. Highly precision numerical algorithms on space-time domain were presented to solve 1+1 dimensional moving boundary problems. An initial moving boundary was given to form an irregular physical domain, and a regular region (a rectangular in Cartesian coordinate system) was chosen to cover the irregular physical domain. The heat equation was numerically computed with a barycentric interpolation collocation method (BICM) on space-time regular region with fixed and moving boundary conditions and initial condition to obtain numerical data in regular region. The data on moving boundary of physical domain were computed with barycentric interpolation. Then, the governing equation of moving boundary was solved with BICM to recover a new moving boundary. Repeat the process, numerical data of temperature and final moving boundary position were given. Numerical examples illustrate effectiveness and accuracy of the method.

Key words: moving boundary problem, barycentric interpolation, space-time collocation method, regular region method, heat conductive equation

CLC Number:

O411.3

WANG Zhaoqing, QIAN Hang, LI Jin. Regular Domain Iterative Collocation Method in Space-Time Region for Moving Boundary Problems[J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2021, 38(1): 16-24.

References

[1] 高小龙. 移动边界问题及其在两个实际背景中的应用[D]. 济南:山东大学, 2014.
[2] 张洁, 华泽钊. 冷冻外科中一维相变问题的数值解[J]. 低温工程, 1999, (4):239-244.
[3] 刘君义, 徐明瑜. 药物控释系统中分数阶反常扩散可动边界问题的精确解[J]. 山东大学学报(理学版), 2003, 38(1):28-32.
[4] 郭白奇, 刘慈群. 有相变的热传导问题[J]. 应用数学学报, 1977, (4):51-57.
[5] JAVIERRE E, VUIK C, VERMOLEN F J, et al. A comparison of numerical models for one-dimensional Stefan problems[J]. Journal of Computational & Applied Mathematics, 2006, 192(2):445-459.
[6] REEMTSEN R, KIRSCH A. A method for the numerical solution of the one-dimensional inverse Stefan problem[J]. Numerische Mathematik, 1984, 45(2):253-273.
[7] JOHANSSON B T, LESNIC D, REEVE T. A method of fundamental solutions for the one-dimensional inverse Stefan problem[J]. Applied Mathematical Modelling, 2011, 35(9):4367-4378.
[8] 周本濂, MURRAY W, 吉新华. 求热导方程移动边界问题近似解的一个方法[J]. 应用数学学报, 1987, 10(1):101-105.
[9] RIZWAN U. An approximate-solution-based numerical scheme for Stefan problem with time-dependent boundary conditions[J]. Numerical Heat Transfer:Part B, 1998, 33:269-285.
[10] RIZWAN U. One-dimensional phase change with periodic boundary conditions[J]. Numerical Heat Transfer:Part A, 1999, 35:361-372.
[11] ESEN A, KUTLUAY S. A numerical solution of the Stefan problem with a Neumann-type boundary condition by enthalpy method[J]. Applied Mathematics and Computation, 2004, 148(2):321-329.
[12] SAVOVI S, CALDWELL J. Finite difference solution of one-dimensional Stefan problem with periodic boundary conditions[J]. International Journal of Heat and Mass Transfer, 2003, 46(15):2911-2916.
[13] LEE T E, BAINES M J, LANGDON S. A finite difference moving mesh method based on conservation for moving boundary problems[J]. Journal of Computational and Applied Mathematics, 2015, 288:1-17.
[14] SADOUN N. On the refined integral method for the one-phase Stefan problem with time dependent boundary conditions[J].Applied Mathematical Modeling, 2006, 30:531-544.
[15] RIZWAN U. A nodal method for phase change moving boundary problems[J]. International Journal of Computational Fluid Dynamics, 1999, 11(3):211-221.
[16] MORLAND L W. A collocation finite element solution for Stefan problems with periodic boundary conditions[J]. Faculty of Sciences and Mathematics, 2016, 30(3):699-709.
[17] WANG L L, JI L, MA W T. Steady heat conduction analysis of functionally graded materials with barycentric Lagrange interpolation collocation method[J]. Chinese Journal of Computational Physics, 2020, 37(2):173-181.
[18] ZHANG B H, YUAN X H, ZHANG Y H, et al. Chebyshev rational approximation method for activation property calculations of radioactive nuclides[J]. Chinese Journal of Computational Physics, 2019, 36(4):457-464.
[19] WEIDEMAN J A C, REDDY S C. A MATLAB differentiation matrix suite[J]. ACM Transactions on Mathematical Software, 2000, 26(4):465-519.
[20] MA H, QIN Q H. An interpolation-based local differential quadrature method to solve partial differential equations using irregularly distributed nodes[J]. International Journal for Numerical Methods in Biomedical Engineering, 2008, 24(7):573-584.
[21] FERNANDES R I, BIALECKI B, FAIRWEATHER G. An ADI extrapolated Crank-Nicolson orthogonal spline collocation method for nonlinear reaction-diffusion systems on evolving domains[J]. Journal of Computational Physics, 2015, 299:561-580.
[22] LANGER U, MOORE S E, NEUMVLLER M. Space-time isogeometric analysis of parabolic evolution problems[J]. Computer Methods in Applied Mechanics and Engineering, 2016, 306:342-363.
[23] HENNIG J. Fully pseudospectral time evolution and its application to 1+1 dimensional physical problems[J]. Journal of Computational Physics, 2013, 235:322-333.
[24] 李树忱, 王兆清. 高精度无网格重心插值配点法-算法、程序及工程应用[M]. 北京:科学出版社, 2012.
[25] 王兆清, 李淑萍. 非线性问题的重心插值配点法[M]. 北京:国防工业出版社, 2015.
[26] 王兆清, 纪思源, 徐子康. 不规则区域平面弹性问题的正则区域重心插值配点法[J]. 计算力学学报, 2018, 35(2):195-201.
[27] 王兆清, 徐子康, 李金. 不可压缩平面问题的位移-压力混合重心插值配点法[J]. 应用力学学报, 2018, 35(3):631-636.
[28] WANG Z Q, XU Z K. Mixed displacement-pressure collocation method for plane elastic problems[J]. Chinese Journal of Computational Physics, 2018, 35(1):77-86.
[29] 王兆清, 张磊, 徐子康. 平面弹性问题的位移-应力混合重心插值配点法[J]. 应用力学学报, 2018, 35(2):304-308.
[30] 李树忱, 王兆清, 袁超. 极坐标系下弹性问题的重心插值配点法[J]. 中南大学学报(自然科学版), 2013, 44(5):2031-2040.
[31] WANG Z Q, QI J S, TANG B T. Numerical solution of singular source problems with barycentric interpolation[J]. Chinese Journal of Computational Physics, 2011, 28(6):883-888.
[32] LIU T, MA W T. Barycentric lagrange interpolation collocation method for two-dimensional hyperbolic telegraph equation[J]. Chinese Journal of Computational Physics, 2016, 33(3):341-348.
[33] WANG Z Q, TANG B T, ZHENG W. A barycentric interpolation collocation method for Darcy flow in two-dimension[J]. Applied Mechanics and Materials, 2014, 684:3-10.
[34] 王兆清, 庄美玲, 姜剑. 非线性MEMS微梁的重心有理插值迭代配点法分析[J]. 固体力学学报, 2015, 36(5):453-459.
[35] JIANG J, WANG Z Q, WANG J H. Barycentric rational interpolation iteration collocation method for solving nonlinear vibration problems[J]. Journal of Computational & Nonlinear Dynamics, 2016, 11(2):1-9.
[36] GVLKAÇ V. Numerical solution of one-dimensional Stefan-like problems using three time-level method[J]. Ozean Journal of Applied Sciences, 2009, 2(1):19-24.