Meshless Collocation Method With Barycentric Interpolation For Solving Helmholtz Equations

In this paper,the two types of meshless barycenteric interpolation collocation methods with the barycentric Lagrange interpolation approximation function and barycentric rational interpolation approximation function,which is employed the Chebyshev interpolation nodes,are established for solving the Helmholtz equations with constant wavenumber and variable wavenumber in the one-dimensional Helmholtz equation to the three-dimensional Helmholtz equation.Firstly,the two kinds of interpolation basis functions are applied to treat the spatial variables and partial derivatives,so the collocation method for solving the second order differential equations are established.Secondly,the discrete algebraic equations of the original equation and the boundary are obtained by substituting the two kinds of barycentric interpolation basis functions into the differential equations on a given node.Then,the differential matrices are used to simplify the given differential equations.Finally,based on the three types of test nodes for the two kinds of interpolation methods,the numerical experiments show that the present method can not only calculate the high wave numbers problems,but also calculate the variable wave numbers problems.In addition,the algorithm has the advantages of good numerical stability,cost-effective and high accuracy etc.First of all,the two kinds of barycentric interpolation collocation formulas are derived for solving the one-dimensional Helmholtz equation.The three types of nodes including the random nodes,Chebyshev nodes and uniform nodes as the interpolation nodes are used to discrete the format.Secondly,the second order differential matrix that corresponding to each interpolation node is obtained.Then the coefficient matrix and the right-hand term of the Helmholtz equation are calculated.Finally,the numerical solution is obtained on the test nodes by substituting the function value,which can be obtained on the interpolation nodes,into the discrete formula.Thereby the numerical errors on the three kinds of test nodes for the two types of interpolation methods are computed.The figures and error figures of corresponding numerical solutions are plotted.The differences between the three types of the test nodes for the two kinds of interpolation methods are compared and analyzed.After then,the two types of barycentric interpolation collocation formulas are derived for solving the two-dimensional Helmholtz equation.The method is used bivariant barycentric interpolation formula,which is more complicated in solving differential matrix and discretization than the onedimensional equation,but the overall idea is the same as it.In addition,numerical solutions for arbitrary fields with irregular boundary shapes are also be obtained,which further illustrate the advantages of barycentric meshless interpolation method.Finally,numerical experiments are conducted to verify the accuracy and stability of the proposed method.In the end,by extending the two-dimensional equation to the three-dimensional equation,the two kinds of barycentric interpolation collocation formulas are deduced for solving the three-dimensional Helmholtz equation.The method uses the multi-variable barycentericic interpolation formula,which is a little more complicated than the two-dimensional equation in terms of solving differential matrix and discretization,but the overall idea is the same as it.Finally,numerical experiments are conducted to verify the accuracy and stability of the proposed method.