The use of the mesh free methods (radial basis functions) in the modeling of radionuclide migration and moving boundary value problems

Recently, the mesh free methods (radial basis functions-RBFs) have emerged as a novel computing method in the scientific and engineering computing community. The numerical solution of partial differential equations (PDEs) has been usually obtained by finite difference methods (FDM), finite element methods (FEM) and boundary elements methods (BEM). These conventional numerical methods still have some drawbacks. For example, the construction of the mesh in two or more dimensions is a nontrivial problem. Solving PDEs using radial basis function (RBF) collocations is an attractive alternative to these traditional methods because no tedious mesh generation is required. We compare the mesh free method, which uses radial basis functions, with the traditional finite difference scheme and analytical solutions. We will present some examples of using RBFs in geostatistical analysis of radionuclide migration modeling. The advection-dispersion equation will be used in the Eulerian and Lagrangian forms. Stefan's or moving boundary value problems will also be presented. The position of the moving boundary will be simulated by the moving data centers method and level set method

Stable PDE Solution Methods for Large Multiquadric Shape Parameters

We present a new method based upon the paper of Volokh and Vilney (2000) that produces highly accurate and stable solutions to very ill-conditioned multiquadric (MQ) radial basis function (RBF) asymmetric collocation methods for partial differential equations (PDEs). We demonstrate that the modified Volokh-Vilney algorithm that we name the improved truncated singular value decomposition (IT-SVD) produces highly accurate and stable numerical solutions for large values of a constant MQ shape parameter, c, that exceeds the critical value of c based upon Gaussian elimination

REGULARIZED MULTIQUADRIC METHOD FOR SOLVING INVERSE BOUNDARY VALUE PROBLEMS

In this paper, we develop a regularized multiquadric method, which is also a non-iterative numerical method, for solving inverse boundary value problems governed by Laplace equation. The well-known ill-posed Cauchy problem is considered, we assume that the boundary conditions are given only on part of the physical boundary of the solution domain, we have to reconstruct the solution and its normal derivative on the rest un-accessible part of the physical boundary. During the whole solution process, we use the multiquadric and the regularization method to construct a regularized multiquadric method. Numerical experiments are given to demonstrate the effectiveness and efficiency of the proposed method