A New Type Finite Difference Method For Acoustic Scattering Problem With High Wave Number

The Acoustic scattering problem is always an interesting and attractive problem,and is also very important research field in the mathematics and physics.These researchs are not only useful in the theoretical,but also meaningful in the application.Such as,in the medical,geophysics,image and so on.Especially,the high wave numbers problems.There are mainly two difficulties in researching this kind of problem: high oscillating of the solution and unbounded computational domain.There are many works focused on this topic.But the “pollution effect” still exists in the high wave number problem in high dimension.In this paper applying the new finite difference menthod proposed in Wang et al.(2015),we investigate the Helmholtz equations with high wave numbers on some special domains,and wish this result were meaningful in researching the numerical method for the acoustic scattering problem.The main work is as follows.1.For the scattering problem in the three-dimensional shallow ocean with an impenetrable circular cylindrical obstacle,firstly,by applying Dt N operator,we transform the unbounded region into a bounded region.Secondly,using separation of variables in the cylindrical coordinate,the three-dimensional problem is transformed into a series of one-dimensional problems.Then using Taylor formula and special the features of the equations,considering the effect of infinite terms in Taylor formula,we construct new finite difference schemes without “pollution effect”.Further,a series of numerical examples are shown to verify the feasibility of this new schemes.2.Based on the above problem results,we study acoustic transmission problems with penetrable obstacles.There are mainly two difficulties when simulating this kind of problem: one is that the wave is mutational on the transmission interface,the other is that the domain is unbounded.For such unbounded transmission problem,we applying the Dt N operator,we are transformed the unbounded domain into bound one,and abtain a scattering problem with artifical boundary.It contains two areas,namely obstacle region and scattering region.We adopt different subdivision in different area,which can not only reduce the computation cost,but also ensure the accuracy of numerical solution on the transmission interface.3.For the problem with high wave number,“pollution effect” can only be eliminate for the problem in one dimension problem,it can not completely eliminate for the high-dimensional problem.But applying higher order finite difference scheme,the simulation accuracy can be improved.Based on the Helmholtz equation with high wave numbers on the annulus,applying the relationship between the reqularity of the solution and the wave number,we wish these results were construct new finite difference scheme to improve the approximated accuracy.