A High Order Finite Volume Method For Solving Helmholtz Equations

Helmholtz equation is often used to describe the scattering and radiation of sound wave and electromagnetic wave as well as the vibration phenomenon of buildings.The solution of such partial differential equation has always been the object of people's attention.The commonly used numerical methods include finite difference method,finite element method,boundary element method and so on.When the wave number is discontinuous or the source term is singular,the solution of the model equation will be discontinuous.The finite volume method is an effective method to solve such problems because it maintains the local conservation of physical quantities and is simple to calculate.In this paper,a compact scheme with high accuracy is constructed for Helmholtz equation with singular source term.The fourth-order compact difference scheme for Helmholtz equation with onedimensional passive term proposed in the previous literature is used to improve the flux discretization process,and a new solution method is obtained.Firstly,the equation is integrated by the finite volume method,and then the flux is expanded by Taylor series,The integral term is approximated by interpolation polynomial.Finally,the equation is discretized by this method.For the one-dimensional problem,firstly,using the method proposed in reference[37],through the improvement of the discrete process,the method is extended to solve the one-dimensional Helmholtz equation with source term,and the wave number continuous and wave number discontinuous conditions are established respectively.Secondly,on the basis of the fourth-order compact format,the discrete method is extended to the sixth-order,mainly by extending the fourth-order interpolation polynomial to the sixthorder interpolation polynomial,a sixth-order compact scheme with continuous and discontinuous wave number is constructed.For the compact scheme of one-dimensional interface problems,it is necessary to deal with both sides of the interface and the interface separately.The continuous scheme is used on both sides of the interface,and the interface scheme is used on the interface.For the interface scheme,the jump condition is used to convert the positive side of the discontinuity to the negative side,and the four point scheme is modified to obtain the three-point compact at the interface.Finally,it is verified by numerical examples.The results show that the format in this paper meets the theoretical accuracy and is suitable for solving large wave number problems.For the two-dimensional problem,based on the one-dimensional problem,the one-dimensional interpolation polynomial is extended to the two-dimensional fourth-order interpolation polynomial and the two-dimensional sixth-order interpolation polynomial,and the four-order mixed compact scheme,the fourth-order mixed compact scheme and the sixth-order mixed compact scheme of the two-dimensional continuous wave number are constructed.Since the right end of the fourth order mixed scheme contains the second derivative of unknown,in order to ensure the overall accuracy of the scheme,the fourth order Pade approximation is adopted for the second derivative,and the boundary scheme with the fourth order precision in reference[38]is adopted.Therefore,the overall accuracy of the six-order mixed compact scheme can reach the fourth order.For the first and second derivatives of the right end of the six-order mixed compact scheme with unknown variables,the algorithm in reference[39]is used to ensure the overall accuracy of the scheme,so that the six-order mixed compact scheme can also reach the theoretical accuracy.Numerical experiments are carried out to verify the results.The results prove the accuracy and validity of the proposed scheme.