Differential Cubature Solution for Helmholtz Equation

摘要

In this paper, In this paper, a novel numerical solution technique, the differential cubature method is employed to solve two-dimensional Helmholtz equations, which are frequently encountered in various fields of engineering and physics. Differential cubature method is a direct discretization method, which approximates a linear operation such as any order of partial derivative of a function or a combination of them by means of polynomials, which expressed as a weighted linear sum of function values at the grid points in the overall physical domain. Since the grid points are arranged arbitrarily in position, this method is capable of solving boundary value problems in various shaped domains. By using this method, Helmholtz equation is transformed into sets of linear algebraic equations about the potential functions at each discrete grid point. Boundary conditions are implemented through discrete grid points on boundary by constraining the values of potential function or the values of derivatives of potential function. Solving this set of algebraic equations, numerical values at each discrete point can be obtained. The convergency, accuracy and applicability of this method are all demonstrated through solving some sample problems of different wave numbers, which have exact solutions. It is concluded that the differential cubature method converged monotonically and very rapidly with increasing the number of grid points for solving Helmholtz equations with various number of waves. By comparison study, we find that the accuracy of present differential cubature method are 32 times greater than the least squares finite element method when the computing scale is the same. It was found that this method is more suitable for solving the Helmholtz equations of lower wave numbers. For Helmholtz equation of higher wave number, much more discrete points are needed to achieve higher accurate solutions. We also find that the accuracy of the present method cannot be improved again when the number of grid points is above a certain value. This is due to the intrinsic property of the differential cubature method by which the matrix for determining the weighting coefficients will be singular when too much grid points are used, since it is based on polynomials.