Numerical study of temperature distribution in an inverse moving boundary problem using a meshless method

In this paper, we consider the inverse one-phase one-dimensional Stefan problem to study the thermal processes with phase change in a moving boundary problem and calculate the temperature distribution in the given domain, as well as approximate the temperature and the heat flux on a boundary of the region. For this problem, the location of the moving boundary and temperature distribution on this curve are available as the extra specifications. First, we use the Landau’s transformation to get a rectangular domain and then apply the Crank–Nicolson finite-difference scheme to discretize the time dimension and reduce the problem to a linear system of differential equations. Next, we employ the radial basis function collocation technique to approximate the spatial unknown function and its derivatives at each time level. Finally, the linear systems of algebraic equations constructed in this way are solved using the LU factorization method. To show the numerical convergence and stability of the proposed method, we solve two benchmark examples when the boundary data are exact or contaminated with additive noises.

[1]  V. Shcherbakov.  Radial basis function partition of unity operator splitting method for pricing multi-asset American options , 2016 .

[2]  Shmuel Rippa,et al.  An algorithm for selecting a good value for the parameter c in radial basis function interpolation , 1999, Adv. Comput. Math..

[3]  Mehdi Dehghan,et al.  Meshless local boundary integral equation (LBIE) method for the unsteady magnetohydrodynamic (MHD) flow in rectangular and circular pipes , 2009, Comput. Phys. Commun..

[4]  B. Blackwell,et al.  Inverse Heat Conduction: Ill-Posed Problems , 1985 .

[5]  Daniel N. Ostrov,et al.  On the Early Exercise Boundary of the American Put Option , 2002, SIAM J. Appl. Math..

[6]  D. Notz,et al.  Insights into brine dynamics and sea ice desalination from a 1-D model study of gravity drainage , 2013 .

[7]  Mehdi Dehghan,et al.  A meshless method for solving nonlinear two-dimensional integral equations of the second kind on non-rectangular domains using radial basis functions with error analysis , 2013, J. Comput. Appl. Math..

[8]  Cheng-Liang Chang.  Pretreatment of wastewater by vacuum freezing system in a cool–thermal storage process , 2002 .

[9]  Steven I. Barry,et al.  Exact and numerical solutions to a Stefan problem with two moving boundaries , 2008 .

[10]  Jamal Amani Rad,et al.  Numerical simulation of reaction-diffusion neural dynamics models and their synchronization/desynchronization: Application to epileptic seizures , 2019, Comput. Math. Appl..

[11]  Holger Wendland,et al.  Scattered Data Approximation: Conditionally positive definite functions , 2004 .

[12]  Hojatollah Adibi,et al.  A meshless discrete Galerkin (MDG) method for the numerical solution of integral equations with logarithmic kernels , 2014, J. Comput. Appl. Math..

[13]  Gregory E. Fasshauer,et al.  Meshfree Approximation Methods with Matlab , 2007, Interdisciplinary Mathematical Sciences.

[14]  Felix E. Browder,et al.  The One-Dimensional Heat Equation: Preface , 1984 .

[15]  Elisabeth Larsson,et al.  Radial basis function partition of unity methods for pricing vanilla basket options , 2016, Comput. Math. Appl..

[16]  J. Crank.  Free and moving boundary problems , 1984 .

[17]  S. Atluri,et al.  The meshless local Petrov-Galerkin (MLPG) method , 2002 .

[18]  B. Tomas Johansson,et al.  A Meshless Regularization Method for a Two-Dimensional Two-Phase Linear Inverse Stefan Problem , 2013 .

[19]  Damian Slota,et al.  Direct and inverse one-phase Stefan problem solved by the variational iteration method , 2007, Comput. Math. Appl..

[20]  Jamal Amani Rad,et al.  Pricing European and American options by radial basis point interpolation , 2015, Appl. Math. Comput..

[21]  D. Needham,et al.  The development of a wax layer on the interior wall of a circular pipe transporting heated oil: the effects of temperature-dependent wax conductivity , 2014, Journal of Engineering Mathematics.

[22]  Jamal Amani Rad,et al.  Numerical pricing of American options under two stochastic factor models with jumps using a meshless local Petrov-Galerkin method , 2014, ArXiv.

[23]  K. Parand,et al.  Kansa method for the solution of a parabolic equation with an unknown spacewise-dependent coefficient subject to an extra measurement , 2013, Comput. Phys. Commun..

[24]  G. Turk,et al.  The use of the mesh free methods (radial basis functions) in the modeling of radionuclide migration and moving boundary value problems , 2007 .

[25]  William H. Press,et al.  Numerical recipes in Fortran 90: the art of parallel scientific computing, 2nd Edition , 1996, Fortran numerical recipes.

[26]  Marcos Antón Amayuelas.  The Stefan problem , 2015 .

[27]  A. Kirsch,et al.  A method for the numerical solution of the one-dimensional inverse Stefan problem , 1984 .

[28]  Jamal Amani Rad,et al.  The meshfree strong form methods for solving one dimensional inverse Cauchy-Stefan problem , 2017, Engineering with Computers.

[29]  Amitesh Kumar,et al.  A novel approach to improve the efficacy of tumour ablation during cryosurgery. , 2013, Cryobiology.

[30]  Saeid Abbasbandy,et al.  Local weak form meshless techniques based on the radial point interpolation (RPI) method and local boundary integral equation (LBIE) method to evaluate European and American options , 2014, Commun. Nonlinear Sci. Numer. Simul..

[31]  Elisabeth Larsson,et al.  A Radial Basis Function Partition of Unity Collocation Method for Convection–Diffusion Equations Arising in Financial Applications , 2015, J. Sci. Comput..

[32]  Damian Słota,et al.  The application of the homotopy perturbation method to one-phase inverse Stefan problem , 2010 .

[33]  Elisabeth Larsson,et al.  Forward deterministic pricing of options using Gaussian radial basis functions , 2017, J. Comput. Sci..

[34]  J. Cannon,et al.  Existence, Uniqueness, Stability, and Monotone Dependence in a Stefan Problem for the Heat Equation , 1967 .

[35]  H. Adibi,et al.  Boundary Determination of the Inverse Heat Conduction Problem in One and Two Dimensions via the Collocation Method Based on the Satisfier Functions , 2018 .

[36]  Guirong Liu.  Mesh Free Methods: Moving Beyond the Finite Element Method , 2002 .

[37]  Guirong Liu,et al.  A point interpolation meshless method based on radial basis functions , 2002 .

[38]  Radoslaw Grzymkowski,et al.  One-phase inverse stefan problem solved by adomian decomposition method , 2006, Comput. Math. Appl..

[39]  M. Dehghan,et al.  Meshless Local Petrov--Galerkin (MLPG) method for the unsteady magnetohydrodynamic (MHD) flow through pipe with arbitrary wall conductivity , 2009 .

[40]  Saeed Kazem,et al.  Radial basis functions methods for solving Fokker–Planck equation , 2012 .

[41]  Amit Saxena,et al.  Computational model of the freezing of jet fuel , 2005 .

[42]  M. Grae Worster,et al.  Desalination processes of sea ice revisited , 2009 .

[43]  G. Smith,et al.  Numerical Solution of Partial Differential Equations: Finite Difference Methods , 1978 .

[44]  Felix E. Browder,et al.  The One-Dimensional Heat Equation: Preface , 1984 .

[45]  Gui-Rong Liu,et al.  An Introduction to Meshfree Methods and Their Programming , 2005 .

[46]  Saeid Abbasbandy,et al.  Comparison of meshless local weak and strong forms based on particular solutions for a non-classical 2-D diffusion model , 2014 .

[47]  Peter Jochum,et al.  The numerical solution of the inverse Stefan problem , 1980 .

[48]  T. Reeve.  The method of fundamental solutions for some direct and inverse problems , 2013 .

[49]  S. Abbasbandy,et al.  Meshless simulations of the two-dimensional fractional-time convection–diffusion–reaction equations , 2012 .

[50]  Guirong Liu.  Meshfree Methods: Moving Beyond the Finite Element Method, Second Edition , 2009 .

[51]  J. Cannon,et al.  Remarks on the one-phase Stefan problem for the heat equation with the flux prescribed on the fixed boundary , 1971 .

[52]  Diego A. Murio,et al.  Solution of inverse heat conduction problems with phase changes by the mollification method , 1992 .

[53]  Elyas Shivanian,et al.  Boundary value identification of inverse Cauchy problems in arbitrary plane domain through meshless radial point Hermite interpolation , 2019, Engineering with Computers.

[54]  E. Kansa,et al.  Solving One-Dimensional Moving-Boundary Problems with Meshless Method , 2008 .

[55]  A. D. Solomon,et al.  Mathematical Modeling Of Melting And Freezing Processes , 1992 .

[56]  R. Witula,et al.  Solution of the one-phase inverse Stefan problem by using the homotopy analysis method , 2015 .

[57]  Chris Chatwin,et al.  Computational Moving Boundary Problems , 1994 .

[58]  Jamal Amani Rad,et al.  Application of meshfree methods for solving the inverse one-dimensional Stefan problem , 2014 .

[59]  S. Atluri,et al.  A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics , 1998 .

[60]  Youssef El Seblani,et al.  Data discovering of inverse Robin boundary conditions problem in arbitrary connected domain through meshless radial point Hermite interpolation , 2020, Engineering with Computers.

[61]  S. Atluri,et al.  The Meshless Local Petrov-Galerkin (MLPG) Method: A Simple \& Less-costly Alternative to the Finite Element and Boundary Element Methods , 2002 .

[62]  Satya N. Atluri,et al.  MESHLESS LOCAL PETROV-GALERKIN (MLPG) METHOD FOR CONVECTION DIFFUSION PROBLEMS , 2000 .

[63]  K. Parand,et al.  Numerical solution of nonlinear Volterra-Fredholm-Hammerstein integral equations via collocation method based on radial basis functions , 2012, Appl. Math. Comput..

[64]  Mehdi Dehghan,et al.  Numerical study of three-dimensional Turing patterns using a meshless method based on moving Kriging element free Galerkin (EFG) approach , 2016, Comput. Math. Appl..

[65]  R. Bhargava,et al.  Simulation of Phase Transition During Cryosurgical Treatment of a Tumor Tissue Loaded With Nanoparticles Using Meshfree Approach , 2014 .

[66]  Daniel Lesnic,et al.  A method of fundamental solutions for the one-dimensional inverse Stefan problem , 2011 .

[67]  William E. Smith,et al.  Product-Integration Rules Based on the Zeros of Jacobi Polynomials , 1980 .

[68]  Jim Douglas,et al.  The Cauchy Problem for the Heat Equation , 1967 .

[69]  Elisabeth Larsson,et al.  A Least Squares Radial Basis Function Partition of Unity Method for Solving PDEs , 2017, SIAM J. Sci. Comput..

[70]  Jamal Amani Rad,et al.  Pricing American options under jump-diffusion models using local weak form meshless techniques , 2017, Int. J. Comput. Math..

[71]  A. M. Meirmanov,et al.  The Stefan Problem , 1992 .

[72]  S. Abbasbandy,et al.  MLPG method for two-dimensional diffusion equation with Neumann's and non-classical boundary conditions , 2011 .

[73]  Elisabeth Larsson,et al.  Radial Basis Function Methods for the Rosenau Equation and Other Higher Order PDEs , 2017, Journal of Scientific Computing.

[74]  Chein-Shan Liu,et al.  Solving two typical inverse Stefan problems by using the Lie-group shooting method , 2011 .

[75]  E. Kansa,et al.  HKBU Institutional Repository , 2018 .

[76]  John van der Hoek,et al.  The one phase Stefan problem subject to the specification of energy , 1982 .

[77]  Saeed Kazem,et al.  A numerical solution of the nonlinear controlled Duffing oscillator by radial basis functions , 2012, Comput. Math. Appl..

[78]  Mehdi Dehghan,et al.  The meshless local Petrov–Galerkin (MLPG) method for the generalized two-dimensional non-linear Schrödinger equation , 2008 .

[79]  Damian Slota,et al.  A new application of He's variational iteration method for the solution of the one-phase Stefan problem , 2009, Comput. Math. Appl..

[80]  M. P. Levin,et al.  Numerical Recipes In Fortran 90: The Art Of Parallel Scientific Computing , 1998, IEEE Concurrency.

[81]  Mehdi Dehghan,et al.  Element free Galerkin approach based on the reproducing kernel particle method for solving 2D fractional Tricomi-type equation with Robin boundary condition , 2017, Comput. Math. Appl..

[82]  Mehdi Dehghan,et al.  Combination of meshless local weak and strong (MLWS) forms to solve the two dimensional hyperbolic telegraph equation , 2010 .