The Compact Difference Numerical Simulation Of One-dimensional Linear Helmholtz Equation

In this paper,we consider the one-dimensional Helmholtz equation as follows:(?),where k is the wave number,f as the source function,u to be said the pressure of the wave field.Helmholtz equation mainly describes a kind of wave propagation phenomena,including electromagnetic wave,sound wave,light radiation and so on.The numerical solution is one of the hot topics in the field of numerical solution of differential equations.For the Helmholtz equation on the bounded domain,the compact finite difference method is used in the high order compact method.The disadvantage of this method is that it will reduce the accuracy of the format and not be conservative when dealing with the non-uniform mesh.Therefore,in the first chapter of this paper,we use the compact finite volume method to study the boundary-value problem of Helmholtz equation in a bounded domain.The first chapter in the second section,we introduce the origin and the physical background of the Helmholtz equation.In the third and the fourth part,we present a compact finite volume method for the one-dimensional Helmholtz equation based on Dirichlet and periodic boundary value problems.For the fifth section,the results of the two experiments show that the proposed schemes are four-order scheme,and the format of the periodic boundary value problem is also applicable to the problem of large wave number.For the Helmholtz equation on the unbounded domain,the unbounded domain leads to great difficulty in solving the problem.At present,the perfectly matched layer(PML)is one of the effective methods to solve this problem.Damping in the layer,the waves do not produce reflections across the inner boundary,and reach the outer boundary of the absorption layer to zero so that we can use the simple homogeneous boundary conditions.Thus,the boundary value problem with artificial boundary conditions on the finite computational region is a good approximation of the original problem.Based on these considerations,in the second chapter of this paper constructs a perfectly matched layer(PML)on the unbounded domain.The second chapter in the second section,we obtain the PML-Helmholtz equation of the Helmholtz equation by using the complex coordinate extension method.Therefore,the Helmholtz equation on the unbounded domain is reduced to the boundary value problem on the bounded computational domain.The finite volume method and the finite difference method are effective methods for solving the boundary value problem of PML-Helmholtz equation.In this paper,the finite volume scheme is proposed in the third section of the second chapter.Good numerical results are obtained,but the accuracy is not high.The finite difference methods often produce serious numerical dispersion,high-order explicit finite difference scheme need to use more grid points,not conducive to the processing of the boundary at the same time.To solve this problem,we use the compact finite volume method and the compact finite difference method to solve the boundary value problem of PML-Helmholtz equation.The two schemes can achieve the accuracy of the four order in the internal domain and artificial boundaries.The compact finite volume scheme is proposed in the second chapter of the fourth section.For the third chapter in the second quarter,we give a compact finite difference scheme for the boundary value problem of PML-Helmholtz equation.The stability of the scheme is proved by the prior error estimate of the difference decomposition.At the end of the two chapter,we give a numerical example to demonstrate the effectiveness of the two schemes.