Research On High Order Hybrid Finite Difference Method For Helmholtz Equation

The Helmholtz equation has been widely used in many engineering practical problems,such as aerospace,ocean engineering,and wave detection.Many researchers at home and abroad have done a lot of research on the numerical solution of Helmholtz equation.Both finite difference method and finite element method It is a classic partial differential equation(PDE)numerical method.It is important to build a finite difference format that is accurate and fast to calculate and effective for large wave number problemsIn this paper mainly constructs a high-order hybrid compact difference scheme for Helmholtz equa-tion.First,based on Taylor series expansion,we get uxx,and the relationship between the difference operator and the differential operator is used to reduce the high-order derivative term in uxx.A sixth-order mixed difference scheme of the one-dimensional(1D)Helmholtz equation is derived.On this basis,the original equation is transformed to obtain the required second-order derivative,which is then derived and the difference operator and differential operator are used again.The relationship between the two 1D Helmholtz equations with six-order precision is constructed.The truncation errors of the three differen-tial formats are analyzed,and use the chasing method to solve the discrete equations.The precision and accuracy of the three formats are verified by numerical examples,and the advantages and disadvantages of the three formats are compared.For the two-dimensional(2D)problem,the sixth-order hybrid difference scheme of the 1D Helmholtz equation is extended to the 2D Helmholtz equation,two kinds of sixth-order hybrid difference schemes of 2D Helmholtz equation are constructed,and solve the discrete equations by using the Biconjugate gradient stabilized(BiCGSTAB)method.The numerical examples are used to verify the precision and accuracy of the two formats,and the advantages and disadvantages of the two formats are compared.