Regular Domain Iterative Method In For Moving Boundary Problems

Governing equation of moving boundary diffusion problem is the heat conduction diffusion equation,whose physical boundary of the defined solution domain changes with time.According to the understanding of the moving boundary information,it can be divided into two categories: one is that the evolution of the moving boundary is known,and the change law of the field variable is solved;the other is that the evolution of the moving boundary is unknown,and the field variable needs to be solved by joint control equation and the evolution equation of moving boundaries.The diffusion problem of the known boundary evolution pattern is a special case of the moving boundary diffusion problem,and its boundary evolution pattern is known in advance.In generally,the defined solution domain of the known boundary evolution diffusion problem is an irregular region in the space-time domain.A space-time regular domain collocation method for numerically solving the initial boundary value problem of the boundary evolution equation is proposed.The irregular defined solution omain is embedded in a regular area(rectangular area),and the diffusion equation is discretized using the barycentric interpolation collocation method on the regular area to obtain the discrete algebraic equation for solving the diffusion equation on the irregular area.The Dirichlet or Neumann boundary conditions on regular and irregular boundaries are discreted by barycentric interpolation,and the discrete boundary conditions are applied to solve the algebraic equations to obtain the numerical solution of the diffusion equation in the regular region,and then the barycentric interpolation is used to obtain the numerical solution in physical domain.The formula of the proposed method is given in detailed,the corresponding calculation program is compiled.Numerical examples verify the effectiveness and accuracy of the proposed method.In the numerical analysis,the barycentric Lagrange interpolation collocation method and the barycentric rational interpolation collocation method are used in the regular area.In the post-processing process,the barycentric interpolation is used to calculate the diffusion function values in the physical region,and the accuracy loss of the interpolation calculation is negligible.The numerical results show that the proposed algorithms have highly accuracy.The accuracy of the Chebyshev nodes is better than the equidistant nodes,and the barycentric Lagrange interpolation is superior to the barycentric rational interpolation.Using equidistant node calculation,when the number of nodes is large,the calculation accuracy of the barycentric Lagrange interpolation collocation method may reduce sharply,but the rational barycentric interpolation collocation method can still obtain very high-precision.The moving boundary problem is a typical nonlinear problem,because the moving boundary evolution pattern is unknown.A high-precision iterative algorithm in the space-time domain to solve the 1+1 dimensional moving boundary diffusion problem is presented.Assuming an initial moving boundary position in the space-time domain,which constitutes the irregular calculation area of the moving boundary problem,an appropriate regular area(rectangular area)is selected to completely cover the calculated irregular area.In the regular area,the moving boundary constraints and fixed boundary conditions are used,and the 1+1 dimensional diffusion equation is solved using the space-time barycentric interpolation collocation method to obtain the data of the field variables in the regular area.Two-dimensional barycentric interpolation is used to calculate the values of the partial derivative of the diffusion function on the hypothetical moving boundary,and then the one-dimensional barycentric interpolation collocation method is used to solve the ordinary differential equation on the moving boundary to obtain a new hypothetical moving interface position.Repeat the above process to finally get the numerical solution of the field variable of the problem and the final position of the moving boundary.The effectiveness and accuracy of presented method are verified by some numerical tests.The numerical results show that the proposed algorithms have highly accuracy.The choice of the location of the initial hypothetical moving boundary has some flexibility,and whether the initial hypothetical moving boundary function satisfies the initial conditions or not,a reliable numerical solution can be obtained in highly precision.For non-smooth discrete data assumed initial boundary positions,highly precision numerical results can also be obtained.In practice,because the position of the moving boundary is unknown,a large regular region can be selected for calculation first,and based on the obtained calculation results,a regular region can be reasonably selected for high-precision analysis.